Algebraic Study on the Quantum Calogero Model

Transcription

Algebraic Study
on
the Quantum Calogero Model
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Thesis
Algebraic Study
on
Hideaki Ujino
Department of Physics, Graduate School of Science,
University of Tokyo,
Hongo 1-3-1, Bunkyo-ku, Tokyo 113, Japan
December , 1996
Acknowledgments
I would like to express my sincere gratitude to my supervisor, Professor i\liki \\"adati , for
having introduced me to the world of quantum integrable systems that has a lot of attracti,·e
open problems. I was really lucky to start my graduate study under his excellent guidance.
He has continuously encouraged me all through the way of my graduate study and has made a
critical reading of the manuscript of my thesis .
I would also like to express my gratitude to Professor Peter John Forrester. He gave us
a seminar on the theory of the Jack symmet ric polynomials in the second year of my graduate study. The seminar played a crucial role in attracting me to the theory of mutivariable
hypergeometric functions.
I appreciate valuable discussions with Prof. Tetsuo Deguchi , Prof. Kiyoshi Sogo, Prof.
Anatol N. Kirillov, Prof. lllasatoshi Noumi, Dr. Takeshi Iizuka, Dr. Taro l\agao, Dr. Frank
Giihmann, Dr. Taichiro Takagi, Dr. Kyoichi Tsurusaki, Dr. Shintaro ll lori , Dr. Kazuaki
Nakayama, Dr. Hidetoshi Awata , Dr. Jun'ichi Shiraishi, Dr. Akira Terai , lllr. Hideyuki
1\lizuta, lllr. i\lasahiro Sh iraishi, lllr. Shuichi lllurakami, i\lr. Toshinao Akuza\\·a, ll lr. Yasumasa Kajinaga and lllr. Yasushi Komori. And I thank all the members of \Vadati group for
sharing a nice and stimulating atmosphere.
I am grateful to the Japan Society for the Promotion of Science and the Aikyo Ikuei-kai for
financial supports.
And last, but not in the least , I thank my parents for allowing me to st udy in the Ph.D.
program.
iii
11
List of Papers Submitted for the Requirement of t he
Degree
Papers Added in References
1. H. Ujino, K. Hikami and l\1. Wadati: Integrability of the Quantum Calogero- Moser i\lodcl,
J. Phys. Soc. Jpn. 61 (1992) 3425-3427.
1. H. Ujino and l\1. \Vadati: Algebraic Approach to the Eigenstates of the Quantum Calogero
l\lodel Confined in an External Harmonic Well, Chaos, Solitons & Fractals 5 (1995) 109118.
2. l\1. Wadati, K. Hikami and H. Ujino: Integrability and AJgcbraic Structure of the Quantum Calogero-:-.!oser l\lodel, Chaos, Solitons & Fractals 3 (1993) 627- 636.
2. H. Ujino and l\1. \Vadati: The Quantum Calogero !\lode! and the TV-Algebra, J. Phys.
Soc. Jpn . 63 (199-1) 3585- 3597.
3. H. Ujino , i\1. \\'adati and K. Hikami: The Quantum Calogero-:-.loser l\!odel: Algebraic
Structures, J. Phys. Soc. Jpn. 62 (1993) 3035- 3043.
3. H. Ujino and 1-.1. Wadati: Orthogonal Symmetric Polynomials Associated with the Quantum Calogero Model , J. Phys. Soc. Jpn . 64 (1995) 2703- 2706.
4. H. Ujino and :-.1. \\"adati: The SU(v) Quantum Calogero l\!odel and the IV00 Algebra
with SU(v) Symmetry, J. Phys. Soc . Jpn. 64 (1995) 39- 56.
4. H. Ujino and M. Wadati: Correspondence between QIS:-.1 and Exchange Operator Formalism , J. Phys. Soc. Jpn. 64 (1995) 4121-4128.
5. H. Ujino and l\1. \Vadati: Exclusion Statistics Interpretation of the Quantum Calogero
Model, J. Phys. Soc. Jpn . 64 (1995) 4064-4068.
5. H. Ujino and M. Wadati: Algebraic Construction of the Eigenstates for the Second Conserved Operator of the Quantum Calogero Model, J . Phys. Soc. Jpn . 65 (1996) 653-656.
6. H. Ujino and 1-.1. Wadati: Thermodynamics of the Exclusion Statistics Associated with
the Quantum Calogero l\!odel, J. Phys. Soc. Jpn. 65 (1996) 1203- 1212.
6. H. Ujino and M. Wadati: Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero l\lodel, J. Phys. Soc. Jpn. 65 (1996) 2423- 2439.
7. H. Ujino and l\1. Wadati: Exclusion Statistical TBA and the Quantum Calogero l\!odel,
J. Phys. Soc. Jpn. 65 (1996) 3819- 3823.
7. H. Ujino and l\1. Wadati: Orthogonality of the Hi-Jack Polynomials Associated with the
Quantum Calogero i\!odel, to be published in J. Phys. Soc. Jpn. 66 (1997) !\"o. 2.
8. H. Ujino: Orthogonal Symmetric Polynomials Associated with the Calogero /-.lode! ,
submitted to Extended and Quantum Algebras and their Applications in Mathematical
Physics, proceedings of the Canada-China l\!eeting in Mathematical Physics, Tianjin ,
China, August 19- 2-1 , 1996, ed. l\1.-L. Ge, Y. Saint-Au bin and L. Vinet (Springer- \"erlag).
Abstract
Among the models in theoretical physics, exactly solvable models deserYe special interest.
For instance, the problem on the hydrogen atom, which played a crucial role in bringing high
credit to quantum mechanics, is a typical example of the models whose eigenvalue problems
are exactly soh·able by separation of variables. This system has three, which is the same as the
number of degrees of freedom of the system, independent and mutually commuting conserved
operators, namely, the Hamiltonian, the total angular momentum and its z-a,is component.
We call a quantum system with N degrees of freedom that has N, independent and mutually
commuting conserved operators quantum integrable system. Important problems for the quantum integrable system are, for instance, the proof of quantum integrability by construction of
the commuting consen·ed operators, the clarification of the underlying symmetries, the exact
solution for the eigem·alue problem and the identification of the simultaneous eigenfunctions
of the commuting conserYed operators. In this thesis, we shall study the quantum Calogero
model, which is a representati,·e model among the one-dimensional quantum particle systems
with inverse-square long-range interactions, from the Yie"·point of the quantum integrable systems.
In Chapter 1, we shall clarify the motive of our study by briefly summarizing the theory of
completely integrable systems in classical mechanics and some difficulty in its quantization. The
completely integrable system is a classical dynamical system with N degrees of freedom that has
N, independent and mutually Poisson commuting conserved quantities. Exact soh·ability by
principle of such a system is guaranteed by Liouville 's theorem on integrability. The definition
of the quantum integrable system is just a translation by the correspondence principle, which
asserts the possibi li ty by principle of the identification of all the quantum numbers. E,·en for
completely integrable systems, it is not trivial to soh·e their initial Yalue problems in practice.
But fortunately, effecti,·e techniques for those problems such as the Lax formulation and the
inverse scattering method ha,·e been de,·eloped for various completely integrable systems. The
Lax formulation was also introduced for the classical Calogero model, which enables constructing mutually Poisson commuting conserYed quantities and soh·ing the initial ,·alue problem for
the model. Howe,·er, due to the non-commutati,·ity between the canonical-conjugate ,·ariables,
the quantized La' formulation does not give a way to construct the commuting consen·ed operators. The aim and contents of the thesis are to give a solution for this difficulty, construct the
conserYed operators, clarify the underlying symmetries, give the exact solution for the eigenvalue problem and identify the simultaneous eigenfunctions of all the commuting conserved
operators.
In Chapter 2, ''"e shall introduce a new formulation for the quantum Calogero model that
corresponds to the Lax formulation in the classical theory, and clarify its quantum integrability
and the underlying symmetry. It was pointed out by Calogero and Olshanetsky et al. that the
,.
\·ii
quantization of the classical Lax equation for the classical model with symmetrizing the noncommutat i,·e products yields an equality for the "quantized classical Calogero Hamiltonian"
wh ich does not contain a quantum correction in t he coupling constant . This observation had
been a ground for the co nj ecture that the quantizat ion of the classical conserved quantities
by the correspondence prin ciple might yield the conserved operators of the quantum Calogero
model. On the other hand , a naive quantization by the corres pondence principle of the classical
Lax equation without symmetrization gi,·es an equali ty for the correct quantum Calogero model
which has a quantum correction in the coupling constant. Howeve r, th ere has not bee n a way
to construct the conserved operators from the quantum Lax equation, let alone a way to prove
their mu t ual commutativity. We pay attention to the sum-to-zero property of theM-matrix in
the Lax equat ion of the qu antum Calogero model , that is, the sum of one of the indices of the Mmatrix is zero. From t his property, we show that the sum of all the elements instead of the trace
in the classical t heory yields the conserved operators of t he quantum Calogero model. ~loreover,
we recursively construct the generalized Lax equations for a fa mily of operators that includes
the conserved operators. We prove that the commutator algebra of the family of operators is
the W -algebra that is the underlying symmetry of the quantum Calogero model. We also give
a way of constructing mutually commuting conserved operators, which proves the quantum
integrability. The Dunk! operator formulation , which was introdu ced by Polychronakos, is also
a powerful method for the study of the quantum Calogero model. We study the relationships
between the quan tum Lax and the Dunk! operator formulations a nd clarify t heir relationship .
The energy eigenvalu e problem of the quantum Calogero model was solved by Calogero. He
derived and solved an ordinary differential equation by separation of variables that determines
the energy eigenvalue. On the other hand , Perelomov paid attention to the close similarity of
the energy eigenvalues of the quantum Calogero model and t hat of the N harmonic oscillators.
He conj ectured that t here must be N independent creation-like operators for the quantum
Calogero model which are proved to be sufficient for the algebraic construction of all the energy eigenfunctions from the consideration of degeneracies. And he succeeded in giving the
first three of such operators. Th e first aim of Chapter 3 is to give the complete answer to the
Perelomov 's conjecture. Using the quantum Lax formulation we introduce in Chapter 2, we
construct all the creation-like operators, which we call the power-sum creation operators. vVe
also show that these power-sum creation operators mutually commute. Successive operat ions
of the power-sum creation operators on the ground state wave fun ct ion t hat is the real Laughlin
wave fun ction yield all the eigenfunctions of the qua ntum Calogero Ha miltonian. Th e second
aim of Chapter 3 is to observe some properties of the simulta neous eigenfunctions of the commuting consen·ed operators by explicitly constructing some of them. From the generalized Lax
equation gi,·en in Chapter 2, we get a matrix representation of the second conserved operator
on the energy eigenfunction basis co nst ructed algebraically in Chapter 3. Diagonalizing the
matrix, we get the simu lta neous eigenfunctions from the eigenvectors. By this method , we get
the eigenvalue formula for the second conserved operators as a conj ect ure and the first seven of
the simultaneous eige nfunct ions. These seven simultaneous eigenfunct ion s are uniqu ely identifi ed by th e eigenvalues for the first and the second conserved operators because they have no
degeneracy. J\loreo,·er, the expansion forms of the abo,·e seven fun ct ions with respec t to the
monomial symmetric polynomials show the triangularity with respect to the dominance order
which is similar to the triangularity for the Jack polynomials. This observation strongly suggests a close relationship between the simultaneous eigenfunctions of t he commuting conserved
operators of the quantum Caloge ro model and the Jack polynomials, and gives an important
hint for the definition of the Hi-Jack polynomials.
. Fo llowi ng the results in Chapter 3, we define the Hi-Jack polynomials and pro,·e that thev are
mdeed the Simultaneous eigenfunctions of all the commut ing conserTed operators of the q-uantum Calogero model 111 Chapter 4. Fmt we prove t hat the Calogero model and the Sutherland
model, whose eigenfunct ions are the J ack polynomials, share the common algebraic structure.
By showmg that the quantum Lax and the Dunk! operator formulation s for the Calogero model
are one-parameter deformations of those for the Sutherland model, we prove that the Hi-Jack
polynomial IS a one-parameter deformation of the Jack polynomial. l\'ext, based on the resu lts m Chapter 3 and the correspondence between the Calogero and the Sutherland models.
we define the Hi-J ack polynomials. The Hi-Jack polynomial is a normalized inhomorreneous
symmetric polynomial that is the simultaneous eigenfu nction of the first and the seco:d commuting consen·ed operators of the Calogero model , whose expansion form with respect to the
monomral symrnetrrc polynomials shows t he t riangu larity in the dominance order. \\'e pro,·e
t hat this defimt1on umquely identifies t he Hi-J ack polynomials. :-.Ioti,·ated bv the result b,·
Lapo in te and Vinet, we prove the Rodrigues formula for the Hi-J ack polynom-ials. Usin rr th~
formul a, we prove their integrality and t he facts that the Hi-Jack poh·nomials are indeed the
simultaneous eigenfun ctions of all the commuting consen·ed operator~ of the Calogero model
and hence that they a re the orthogonal symmetric polynomials with t he weight function gi,·en
by the sq uare of the norm of t he gro und state wave function. Consequently, we prove that
the Hr-Jack polyno mial is a multivariable generalizat ion of the Hermite polynomial, which \\·as
mtroduced by Lasselle and J\lacdonald from the viewpoin t of a deformation of an orthorronal
polynom ial t hat co rresponds to the specia l case of the Hi-Jack polynomial. However, it ~ the
first time to deri,·e it from the viewpoint of t he simultaneous eigenfunction of all the commuting
conserved operators of the quantum integrable system, which is a natural object for physicists·
mterest.
As we have described abo,·e, we shall study the quantum Calogero model from the viewpoint
of the quantum in tegrable systems. And we shall deri,·e the results listed below:
• A new formulati on named quantum Lax formula tion that o,·ercomes the difficulty in the
quantization of the classical Lax formulation.
• Const ru ction of t he com mu t ing consen·ed operators a nd the proof of the quantum integrability.
• Derivat ion of the underlying ll'-sym metry.
• Completion of the algeb raic construction of the energy eigenfunction.
• Identificat ion of the simul taneo us eigenfunctions of the commuting consernd operators,
their orthogonal ity a nd t he Rodrigues formula.
Th ese are the main resu lts of the thesis.
Contents
Acknowledgments
List of Papers Submitted for the Requirement of the Degree
Papers Added in References
ii
iii
Abstract
v
1 Introduction
1.1 Integrable Systems
1.2 Calogero !\lode! ·
1.3 !\lotivation and Aims
1
2
8
13
2 Algebraic Structure of the Calogero Model
2.1 Quantum Lax Formulation ·
2.2 W-Symmetry of the Calogero !\lode!
2.3 Dunk! Operator Formulation ·
2.4 Relationship between the Two Formulations
2.5 Summary
15
16
3 Algebraic Construction of the Eigenfunctions
3.1 Perelomov's Program
3.2 Simultaneous Eigenfunctions ·
3.3 Summary
41
42
45
4 Orthogonal Basis
4.1 i\lodels and Formulations ·
4.2 Hi-Jack Symmetric Polynomials
4.3 Proofs ·
4.3.1 Unique Identification of the Hi-Jack Polynomials·
4.3.2 Hamiltonian
4.3.3 Nu ll Operators
4.3.4 Raising Operators ·
4.3.5 Normalization, Triangularity and Expansion
4.3.6 Orthogonality
4.4 Summary
51
52
29
30
32
39
49
60
65
65
69
71
83
92
97
100
C01\"TENTS
X
5
Summary and Concluding Remarks
Bibliography
103
107
Chapter 1
Introduction
Integrable systems haw been playing a special role among many models that appear in
theoretical physics. Simple and solvable models have provided not only good exercises for students, but also excellent ll"ays for us to understand physical phenomena. For instance, the exact
solution of the quantum Kepler problem gaye us a convincing answer to the mystery of the series
of discrete spectra of the solar ray at the beginning of this century [60 , 6-l, 65], which brought
high credit to quantum mechanics. In the analysis, the theory of the Laguerre polynomials
and the spherical harmonics played an important role. These special functions are obtained
by means of separation of ,·ariables of the Hamiltonian . By separation of variables, the partial
differential equation that gi,·es the eigenvalue problem of the model is decomposed into three
ordinary differential equations ll"hose solutions are the Laguerre polynomials and the spherical
harmonics. From the Yie11·point of the integrable systems, we should note that the model has
three, independent and mutually commuting conserved operators, namely the Hamiltonian, the
total angular momentum and its z-a:xis component. These special functions are associated with
the joint wave functions for the three mutually commuting conserved operators of the model.
Thus we can identify all the quantum numbers the model has. Besides the quantum Kepler
problem, the theory of special functions haYe been helping the analysis of Yarious kinds of
important Schriidinger equations since the early days of quantum mechanics. Various physical
phenomena haw been explained by reducing the problems to integrable Hamiltonians as first
approximations and expanding the correction terms.
Apart from the aim to11·ard the explanation of physical phenomena, mathematical structures
of integrable systems attract many researchers because of their beauty. l\Jathematical beauty of
integrable systems moti,·ates me to study them. In classical mechanics , the theory of the completely integrable system is ll"ell developed. The completely integrable system is a dynamical
system ll"ith N degrees of freedom that has N, independent and mutually Poisson commuting
conserved quantities. Liou,·ille 's theorem on integrability guarantees that ll"e can soh-e the
initial ,·alue problem of the completely integrable system by definite integrals and change of
variables in principle. Thus construction of commuting conserved quantities has been an important problem in t he theory of classical integrable systems. The Lax formulation provides a
method to construct consen·ed quantities. The success of the ill\·erse scattering method in the
soliton t heory [1, 28] benefited from the Lax formulat ion. Though there is no counterpart of the
Liouville theorem in quantum mechanics, ll"e define the quantum integrable system in a similar
fashion to the definition of the classical completely integrable system. i\amely, the quantum
integrable system is a quantum system with N degrees of freedom that has N , independent
CHAPTER 1. INTRODUCTIO.\"
2
and mutually commuting consen·ed operators. Existence of such conserved quantities asserts
the possibility of identification of all the quantum numbers of the system. Thus the notion
of integrabili ty and construction of commuting conserved operators a re also im po rtant in the
quant um theory. However, na ive quantization of the classical Lax formulat ion does not work
because of the non-commutativity between t he canon ical-conjugate variables. Du e to t he a bo,·e
difficulty, development of a formulation for t he quantum theory that is the counterpart of the
classical Lax formulation is an attractive and challenging p rob lem.
The main theme of my thesis is the algebraic study on the quantum Calogero model. The
Calogero model was introduced by F. Calogero in 1971 [19], "·hich describes N quantum ident ical particles on a line with inverse-square interactions confined in a n external harmonic well . He
solved the eigenvalue problem for the model by separation of variables. In the cl assical theory,
t he Calogero model is a completely integrable system and has a Lax formulation [52 , 57, 58).
We shall introduce the quantum Lax formulation a nd the Dunk! operator formulation for t he
Calogero model that is the cou nterpart of the classical Lax formulat ion . Our formulati ons will
clarify the conservation law, the a lge braic structure and the s pecial functions that are assoc iated with th e eigenvalue problem of the Calogero mod el. Namely, what we shall show is an
old-fashioned study on the exactly solvable problem in quantum mechanics and the associated
special funct ions by using brand-new tools.
In this chapter, we shall review what is the integrable systems a nd an overview history of the
Calogero model so that t he motive for the problems we shall study in the succeed ing chapters
will be clear. First, we shall summarize the Liom·ille t heorem and the Lax formul ation . \\'e
shall also present the definition of quantum integrability a nd difficulties in the quant ization of
t he Lax formul ation . Summarizing t he overview history and t he classical integrabi lity of the
Calogero model, we shall observe the effectivity of the classical Lax formulat ion in t he study
of t he classical Calogero model. And last , we summarize the motivation and aims of the study
and brief t he outline of the thesis .
1.1
Integrable Syst e m s
Let us first consid er the not ion of integrability in the cl ass ical mechanics. Generally speaking, we can not solve the initial value problem for dyn amical systems with more than or eq ua l to
t\\·o degrees of freedom. We can deal with the interactin g two-body problem whose interaction
only depends on the distance between the particles , for we can separate vari ables by introducing center of mass coordin ates and rel ative coordinates. Thus we usually say t hat many-body
problems involving more t han or equal to three particles are not ge nerally solvable.
Howe,·er, there is a class of dynamical syste ms wh ose initial ,·alue problems are, in prin ciple, soh·able. We call such a classical dynamical system completely integrable system [2).
Com pletely integrable systems a re dynamical systems with N degrees of freedom t hat have 1\' ,
independent a nd involutive , or in other words, mu t ually Poisso n commuting consen·ed quantities. Liouville's t heorem on in tegra bility guarantees that completely integrable syste ms ca n be
solved by quad rat ures. \Ve sha ll verify the theorem and see how it works.
Let us state again Liouville's t heorem on integrability.
1.1. INTEGRABLE SYSTEMS
3
{Pi , PJ}p
= {q, , qi}p = 0,
{q, , p1 }r
= 6, 1,
(llb)
where {a , b} P is the Poisson bracket,
{a, b}p ~r L.N ( aa_!!!!._. - [)b. IJa).
•=I IJq, IJp,
IJq, IJp,
If the system has a set of N, independent and analytic conserved quantities {I;Ji
di,
dt =
{
}
I ,, H r
(llc)
= 1, . .. , N} ,
[)I
+ at' = 0,
det(~~~) ,CO,
(l.2a)
(l.2b)
which are in involution,
{hiJ}p
= 0,
(l.2c)
then the initial value problem of the equation of motion of the system can be solved by quadr _
tures.
a
Key to the proof of Theorem l. 1 is to show the ex istence of a canon ical transformation,
(p ; q) --> (P ; Q), P1 = I1, j
= 1, · · ·, N.
(1.3)
Since the conserved quantities are independent (l.2b) , we can solve a set of equations,
Pi= I1 (p ;q;t) , j = 1, ... , JV,
with respect to p:
Pi= Fi(q ; P ; t), j
= 1, · · ·, N.
(1.4)
Here Fi(q ; P ; t) is an a na lytic fun ction , whose existence is guaranteed by the inverse function
theorem. Computmg partial deri,·at ives of both sides of eq. (1.4) with respect to p1. and q1. we
get
'
(l.5a)
(l. 5b)
T hen the product of eqs. (1.5) yields
L
{\'
[)Fj IJ?t [)Fk IJP, = - [)Fj
i,l,m=l [}f>t IJqi IJPm [}pi
IJqk
(1.6)
.-\ntisym met rizin g eq. (1.6) "·ith respect to t he indices j and k, we ha,·e
IJFk _ IJF1
IJqj
IJqk
Theorem 1. 1 Consider a Hamiltonian with 1·espect to coordinates q and momenta p that
describes a dynamical system with N degrees of freedom,
H(p;q;t) = H (pt. · · · ,p,,·;q 1 , • • · , q,v; t) ,
(l.l a)
(17)
CHAPTER 1. INTRODUCTION
4
The last equality of eq. (1.7) comes from the mutual commutativity of the conserved quantities
with respect to the Poisson bracket (l.2c).
Now we define a function IVr(q, q(i); P; t) by the following integral:
N
IVr(q, q(il ; P; t) =
2:=
k=1
lr
Fk(q ; P ; t)dqk.
1.1. INTEGR..l.BLE SYSTE.\IS
we can calculate the Poisson brackets among the action-angle variables (P; Q) as follows:
{P;,Pi}p
(1.8)
In the aboYe expression (1.8), r means a path inN dimensional coordinate space that connects
two points q and q<iJ. \Ye can see that the above integral is independent of paths r. Let rand
r' be t'm different paths that connect q and q<'J. The difference between 1Vr and Wr• 1s
5
{ Q; , Qj} p
{hijL = o,
E((f)~k
~~
lp;q;t
lq;P) (
O~k ~~
-(_?_I
f)lVI
)(aqk
_?_I
apk
aP;
p;q;t
N
2:=
l,m=l
f)2W
oPoR
q;P;t
I
p;q;t
lq;P)
~1
aP q;P)
1
N (
I q;P;t
i
lp;q;t
)
f)2JV
I
{F\, Pm} P + 1=2:=1 t5il-aPia}\ q;P;t
-81--1 )
f)2W
1
oP;o}\
q;P;t
0,
£((_?_1
ewl
aqk
aP;
k=1
where Sis a hypersurface whose boundary aS is given by the closed path, aS= f - f'. Then
using the Stokes' formula , we get
p;q;t
q;P;t
)aP1
apk
1 _aP;I
(_?_I
ewl
apk
aqk
aP
p;q;t
p;q;t
p;q;t
I + 2:=N -f)2W- I {Pt, Pi}
-f)pj
oP; q;P;t
= oP;o}\ q;P;t
P
1 q;P)
)
1 1
t5ij.
The last equality follows from eq. (1.7). Thus we see the integral (1.8) is independent of paths
r. We denote the path-independent function by
(1.9)
We can easily verify that this function W(q, q(i); P; t) is a generating function of the canon ical
transformation (1.3) in the followin g way. We define the angle variable Q that is canonically
conjugate to to the action variable P by
ew
Qi =
1, · · ·, N.
(1.10)
1
From the definition of the generat ing function (1.9), we can readily see
f) IV
f)f{
dQj
dt
dP1
dt
(l.12a)
aP/
f)J(
- oQi = 0, j = 1, · · ·, N,
j
= 1, · · · , N.
I<= H +oW
lp;q;t
lp;q;t
(1.12c)
The second equation of the canonical equation of motion (1.12) is equivalent to eq. (l.2a)
because P1 is a conserYed quantity 11.
Equation (l.12b) means that the Hamiltonian J( is independent of the angle ,·ariables Q ,
K(P; Q; t) = I<(P; t).
(1.11)
Since the angle ,·ariables P are inYariant under the time evolution, we can fix them at
Paying attention to the independent Yariables of the partial derivati,·es,
f)~k
f)~k
(1.12b)
where the new Hamiltonian I< is given by
at ·
f)P , j =
..,.= Fi =Pi,
UQj
Thus we have \-erified that the transformation (1.3) preserves the fundamental Poisson bracket
(l.1b) by using properties of the generating function (1.11) and hence it is a canonical transformation.
The time Holution of the action-angle variables is described by the canonical equations of
motion,
p (OL
(113)
This means that the Hamil tonian is an analvtic function of time t with N initial parameters
p (o) Substituting eq. (1.13) into eq. (l.12a)" and integrating it overt, we get
Q1·(t) =
Q(O)
1
0 -J<(p(O) · · · p(O). t')dt'
+ j {' -f)p(O)
1 '
'
N '
·
0
1
6
Thus we ha,·e solved the initial value problem in the action-angle variables. We can transform
the action-angle variables and their initial valu es back to the original coordinates and momenta
and their initial values through the canonical transformation (1.3) defined by the generating
function (1.9). So dynamical systems satisfying the assumptions of the theorem can be solved
by quadratures.
A short comment might be in order. Dynamical systems that satisfy the assumptions
of Liouville's theorem on integrability are called completely integrable systems . Liou,·ille's
theorem says that there exists a canonical transformation that decomposes the system with
N degrees of freedom into N systems "·ith only one degree of freedom , whose time evolution
equations we can integrate. However, it does not provide us an explicit form of such a canonical
transformation and its inverse. This means that Liouville 's theorem does not provide us a
practical way to soh·e completely integrable systems. For practical methods to soh·e classical
integrable systems, see, for instance , refs . [1, 28, 67 , 68).
Because of Liouville 's theorem , it has been an interesting problem in classical mechanics to
prO\·e the integrability of a system by showing a systematic way to construct a set of conserved
quantities in in volution. Historically, La.x provided such a method for the system with infinitely
many degrees of freedom [47) . Suppose there is an N x N matrix L of coordinates q and
momenta p . Then the time evolu tion of the matrix L governed by a Hamiltonian H is given
by
dL = {L
+ 8L
dt
'
r
at
H}
1.1. INTEGRABLE SYSTE.\IS
7
To pr~ve the involuti';ity o~ th e conserved quantities (l.2c) , we usually use the classical rmatnx, which IS an N x N matnx that satisfies th e followin g fundam ental Poisson bracket
[28),
{ £(1) ) £(2)} p =
12 , L(llJ- [r 21 , £(2)] .
(1.16)
2
[r
The undefined symbols in the above equation are
L(l l ~£ ® 1, L< 2 l ~ 1 ®L
r21
~ Pr12P, P~
N
:L Eij ® Eji,
{L,TrL"}r
Tr2 { £(1), (L< 2 l)" } r ·
Here , the symbol Tr2 mean~ the trace with respect to the second space of the tensor product,
I. e., Tr2A ® B = A(TrB). Using eq. (1.16) , we have
Tr2{ L(l l, (L<2l)"} r
Tr2 t(£(2l)k-I{L(I l,£(2J} (£(2Jt-k
(114)
k=l
Tr2 ~(£(2J )k-I(h
where the bracket in the r.h.s. denotes the commutator between matrices,
= Li\I- ML.
dL"
p
2 , £(1lj-
b 1, £(2lj)(L(2J)n - k
[L(I l, -nTr2ri2(£(2Jt-I].
We call eq. (1.14) La.x equation. A pair of matrices, L and M, are called the Lax pair.
The Lax equation provides us a systematic way to construct the conse rved quantities of the
Hamiltonian. From the Lax equation (1.14), we ha,·e the equation of motion for the n-th power
of the £-matrix:
dt
= y ® x,
where 1 and Eij are the unit matrix and the matrix unit , 1kt = .5k1, (Eij)kt = .Sikbjt, respectl\·ely.. From eq. (1.16) , we can prove the mutual Poisson commutativity among the conserved
quantities (1.15) as follows. \\"e shall calculate the following Poisson bracket:
We suppose there is another N x N matrix M of coordinates and momenta that satisfies
[L,Mj
Px ® y
i ,j=l
= [L" , M].
Thus we obtain
(1.17)
This is nothing but the Lax equation for the higher consen·ed quantity I., that guarantees the
Poisson commutati,·ity with other consen·ed quantities. Using eq. (1.17) , the Poisson bracket
among the consen·ed quantities In and Im is calculated as follows:
Calculating traces of both sides of the abo,·e equation , we ha,·e
{TrLm,I"}P
iTrL"
dt
Tr[L" , Mj
Tr[Lm , JI.. J
Tr(L"i\1- J\IL")
0.
0,
due to the identity, TrAB = TrB.-1 , for arbitrary c-number ,·alued matrices, A and B. Thus we
can obtain conserwd quantities of the Hamiltonian H just by calculating traces of the powers
of the £-matrix:
I.~TrL" , dt
din
= 0,
n
= 1, 2, · · ·
(1.15)
Thus we conclude that the im·oluti,·ity of the conserved quantity (1.2c) follows from the classical
r -matrix and the fundamental Poisson bracket (1.16). The Lax formulation and the classical
r-matrix are the sta nd ard formulation for the classical integrable systems.
Though there is no theorem in quantum theory that corresponds to Liou,·ille 's theorem in
the classical theory, we often define the quantum integrable system in a simi lar fashion to the
definition of the completely integrable system. The quantum integrable system is a quantum
CHAPTER 1. INTRODUCTIO.V
8
1.2. CALOGERO MODEL
9
system with N degrees of freedom that has N, independent and mutually commuting conserved
where the momentum Pi is given by a partial differential operator, p1 = -i 8 .. Throughout this
operators,
thesis, we set the Planck constant at unity, li = 1. Two parameters, g and w are the coupling
constant and the strength of the external harmonic well, respecti,·ely. In recent years, onedimensional quantum systems with inverse-square interactions including the Calogero model
enjoy rene\\·ed interests of theoretical physicists. Japanese researchers have also contributed
toward attracting many researchers who are interested in the long-range effect of electronelectron interactions in condensed matter to the inverse-square interaction models. .\mong
such contributions, Kawakami and Yang's study on the low-temperature critical behaviour of
the Sutherland model [38] and introduction of the supersymmetric long-range t-J model by
Kuramoto and Yokoyama [40] should be noted.
8 XJ
din - 0 [I I ]
dt- ,
r11
m
=0
'
n m
'
=1
1 2, ... , N,
(1.18)
•
where the time evolu tion of the conserved operator is governed by the Heisenberg equation of
motion,
. din
-1/idt
= [H,Jn l -
.
Bin
1/ifit.
suall we deal with operators that do not explicitly depend on timet. In this case, the second
. y, tl 1 e r h s of the above Heisenberg equation is zero. Existence of mutually commuting
~~~:e~~ed op~r~~ors means that we can specify, in principle, all the quantum numbers the
tern has by identifying the simultaneous eigenfunctions of the consen·ed operators. Thus, the
sys
.
t
t" n of the conserved operators and the identification of their simultaneous
systematic cons rue IO
.
I · t b
ei enfunct ions are important problems in the theory of quantum mtegrable system. t IS o e
n~ted that the construction of the simultaneous eigenfunctiOns for the commutmg conserved
erators in practice and the proof of quantum integrability are different problems, winch IS
op
1 at similar to what the integrability in Liouville's sense means. However, the beautiful
::~ ~vn~ry for th e classical integrable systems such as the Lax formulation and the classical1·matrix can not be applied to the quantum systems because of the non-comm utativity between
the c~nonical conjugate variables. For exam ple, the trace formula for the conserved quant1 t1e~
(1. 15) does not work in the quantum theory. Suppose that the natural quantizatiOn of the Lax
equation holds:
U
From the ,·iewpoint of the classical integrable systems, the classical Calogero model is a
completely integrable system in LioU\·ille's sense that has a Lax formulation [52, 57, 58] and
a classical r-matrix [6]. The idea to introduce the Lax formulation for the inverse-square
in teract ion models is due to i\Ioser [52]. Though i\Ioser dealt with the Calogero-t·doser model,
1
dL
-iii-=
dt
. BLat [L,i\1.l
[H ' Ll -IIi-=
(120)
and the Sutherland model [72, 73],
(121)
Then the time evolution of the power of the £ -matrix is expressed as
d£n
- ihdt = [Ln,
Mj.
However we cannot concl ude from the above equation that the trace of the power of the £matrix i~ a conserved operator because the trace identity, TrA.B = TrBA, does not work for
operator-valued matrices A and B, i.e.,
-irtftTrLn = Tr[Ln, M] ,1 0.
Thus we have to de,·elop some extra device for the systematic analysis of the quantum integrable
systems.
1.2
Calogero Model
The Calogero model was introduced as a solvable quantum many-body mod el in 19 71 [19].
The Calogero model describes N identical particles on a lin e with inverse-square long-range
interactions confined in an external harmonic well ,
1N?1N
1
He=?
Pj + 29
(x _ x .)2
L
-]=I
L
J,k=I
jtk
J
k
l2N2
xi,
+ ?w
-
L
]=I
the Lax formulation for the Calogero model (1.19) is st ra ightforwardly derived from the Lax
formul ation for the Calogero-.\ loser model (1.20). The Sutherland model (1.21) is defined by a
general izat ion of the Calogero-.\ Ioser model (1.20) that is compatible with the periodic boundary condition. In Chapter 4, we shall see a brief summary on t he commo n algebraic structure
of the Calogero and the Sutherland models. The classical Lax formulation was extended further to those for more generalized models, e.g. the ell iptic Calogero-i\loser model [21] and
the inverse-square interaction models associated with the root lat t ices of Lie algebras [58]. At
that time, the integrability of the models were claimed just by showing the La' formulation,
though the Lax formulation is not sufficient to prove the involuti,·ity, or in other words, the
mutual Poisson commutati,·ity among the conserved quantities. Proof of in\"Olutivity for the
classical Calogero model was completed by constructing the classical 1·-matrix [6] . Here we
shall briefly summarize a deri,·ation of t he Lax formulation and the r-matrix method for the
classical Calogero model.
Following the projection method [57, 58], we shall first in troduce the classical Lax formulation for t he Calogero model. The merit of the projection method is that it gives t he general
solu t ion for the initial Yaluc problem of the classical Calogero model. Let us consider a timedependent N x N Hermi t ian matrix X(t). We s hall deal with a case that each element of the
matrix X(t) obeys the equat ion of motion for the harmo ni c oscillator:
(1.19)
(1.22)
CHAPTER 1. INTRODUCTI01\.
10
The general solution of the above equation of motion are readily obtained as
ldX
(1.23)
X(t) = X(O) coswt + ;::;Cft(O) sinwt ,
where X(O) and
~ (0)
1.2. CALOGERO MODEL
Equations (1.26) and (1.27) describe the motion of the eigem·alues of the time-dependent Hermitian matrix X(t). From th e L-and J\1- matrices that satisfy eqs . (1.26) and (1.27) , we can
reproduce the free harmonic oscillation of the Hermitian matrix X(t). For a given J\1-matrix ,
the unitary matrix U(t) that satisfies eq. (1.26c) is obtained as follows:
are the initial coordinate and velocity. We note that the matrix,
U(t)
C(t)~[X(t) , d;~ (t)j,
X J
+ [X(t), d-dtz
(t) = 0,
(1.24b)
(1.24c)
· · matnx
· · X(t) b)· the "projection" or
We consider the motion of eigenvalues of the Hermtttan
an unitary transformation:
Xt(t)
X(t)
o
= U(t)D(t)U- 1 (t),
D(t)
xz(t)
=
·.
(1.25)
[
dU(t)
dt
( U(O)
M(t)
lo
+ [M(t), D(t)j ,
(1. 26b)
~ u- 1 (t)~~ (t).
(1.26c)
1
fa''-•
dt,_ 2
···fa''
(1.28)
dt 1 J\J(tJ)J\1 (tz) · · · M(tt-t)) M(t)
U(t).\!(t).
Thus we want to find out the La' pair, L and J\1, which satisfies eqs. (1.26) and (1.27) with
the time evolution defined by the classical Calogero Hamiltonian.
The number of t he initial parameters of the harmonic oscillation of the Hermitian matrix
X(t) is 2N 2 whereas those of the N-body Calogero model is 2N. Thus we ha,·e to introduce
a restriction to the initi al parameters of X(t) and reduce its number to 2N. As the first
restriction, we restrict the initial ,·alue of matrix X(t) a diagonal matrix:
X(O) =
xz(O) ..
[
XN(O)
j=
D(O).
U(O) = 1
(1.30)
(1.31)
dt(t) + [DI(t), ~(t)U(t)j
dD (t)
dt
L(t) ,
(1.29)
This yields the restriction to the initial value of the unitary matrix ,
dU- 1
dDI
J1!1(t)
M(tt).
The second one is the restriction to the conserved matrix,
Note that the L- and J\1-matrices are respectively Herm it ian and anti-Herm itian ,
Ll(t)
· ··
Xt(O)
dU (t)D(t)u-t(t) + U(t) dD (t)U-t(t)- U(t)D(t)u-t (t) ~~ (t)u-t(t)
dt
dt
U(t)L(t)U- 1 (t),
(1.26a)
L(t) ~ dD(t)
dt
lo
~fa' dt,_
Calcu lating the time-derivative of both sides of the above equation , we have
dX (t)
dt
dt, ('' dtt-t · · · ('' dt 1 J\J(tJ)J\J(t 2 )
(1.2-!a)
2
[-(t), -(t)j
dt
Cl(t) =
l
\Ve can confirm that the abm·e integral is indeed a solution of eq. (1.26c) by the following
calcu lation:
dX
dX
dt
dt
-C(t).
=
= U(O) f
l=O
is an anti-Hermitian conserved matrix:
dC (t)
11
which yields the restriction to the initial velocity of the matrix X(t):
+ [u-t (t) dU (t) , D(t)j
dt
du-t (t)U(t)
dt
dU
-u- 1(t)-(t)
dt
-i\f(t).
(1.32)
Using the above L- , i\f- and D-matrices, the equation of motion for the Herm itian matrix (1.22)
is cast into the following form:
~~ (t)
=
[L(t), .\J(t)j- w2 D(t).
(1.27)
This observation suggests that the form of L-matrix is gi,·en by
(1.33)
1.3. 1\IOTN.l.TION AND AIMS
13
12
.
.
d .. t. . by the Poisson bracket with the classical Calogero Hamiltonian
Definmg the t1me- erna 1ve
(1.19),
dL (t) ~ { L(t), He} p '
dt
we can determine the explicit form of the M-matrix by the sufficient condition of eq. (1.26),
N
r12
( M(t) ) ij
L
1
{
N
- - - Eik 181 L(LikEii
Xk
+ LiiEkt)
I= I
1
N
+ 2Eii 181 L(LijEik + LkiEjl-
LlkEij- LjtEkl)
1=1
< ~
= •tau;i
~ (x;(t)
1
_ x1(t))2
- ia(1 -
H'1
1
(x;(t)- x1(t))
2'
where the coupling constant is given by
(1.35)
· · an d an t.1- Hermitian · They also satisfy
The above L-and M- matrices are respectively Herm1t1an
eq. (1.26) under the identification
dD (t) ~ {D(t), He} p·
Thus we conclude that the general solution for the classical Calogero model (1.19) is obtained
·
by d.1agona1·lZlng
· the time-dependent matrix· (1.23) w1th 101t1al values, eqs. (1.29) and
JUSt
(1. ~~nserved quantities of the Calogero model are obtained by the trace formula (1.15).
introduce new matrices £± by
\\'e
(136)
Then eqs. (1.26) and (1.27) are cast into the following forms:
{L±,He}P = [L±,Mj ±iwL±.
(137)
We call the abo,·e equation Lax equation for the Calogero model. It is straightforward to deri,·e
the following equation from the Lax equation,
(138)
Thus we can obtain the conserYed quantities of the Calogero model by the following trace
Tr(L + L -y\
(1.39)
and the Hamiltonian corresponds to the first conserved quantity,
~~lassical
= 2He.
IIn·oluth·ity of the consen·ed quantities is Yerified by the classical r·-matrix [6]. The fundamental
Poisson bracket of the matrix L + L- are expressed as
(1.40a)
(Eik- Eki) }·
(1.40b)
Thus we confirm the integrability of the classical Calogero model.
\\'e have obserYed that the Lax formulation gi,·es a powerful method to study the classical
Calogero model. The Lax formulation gi,·es not only a way to proYe the integrability, but also
a way to solve the initial Yalue problem of the Calogero model. HoweYer , as we hm·e mentioned
in the previous section, non-commutati,·ity between the canonical-conjugate ,·ariables in the
quantum theory depri,·es the Lax formulation of its merits. So we have to inYent a new device
for the systematic study of the quantum Calogero model.
1.3
dt
3
+ ~Eii 181
(1.3~)
l;ti
J~lassical =
-
i#
[L(t), jH(t)j- w D(t)
=>
=
j,k=l Xj -
2
{L(t) , He} P
formula,
where the classical r-matrix is giYen by
Motivation and Aims
We have briefly summarized what is the integrable systems and how effecti,·ely the Lax formulation works for the classical Calogero model. It is quite natural to expect that the quantized
classical integrable systems might be the quantum integrable systems in a sense they possess
mutually commuting consen·ed operators. It was pointed out by Calogero and Olshanetsky et
al. [20, 59) that the quantization of the classical Lax equation for the classical model with symmetrizing the non-commutative products yields a equality for the "quantized classical Calogero
Hamiltonian" which does not contain a quantum correction in the coupling constant. This observation had been a ground for the conjecture that the quantization of the classical consen·ed
quantities by the correspondence principle might yield the conserved operators of the quantum
Calogero model. On the other hand, a nai,·e quantization by the correspondence principle of
the classical Lax equation without symmetrization gi,·es a equality for the correct quantum
Calogero model which has a quantum correction in the coupling constant. However, the La.x
formulation, which is a precision machinery for the classical integrable systems, does not work
in the quantum theory because the non-commutatiYity between the canonical-conjugate variables breaks the trace identity. Because of this difficulty, a proof of quantum integrability does
not follow from classical integrability. The fact moti,·ates us to find out a new instrument for
the quantum integrable systems. In this thesis, we shall try such a challenge for the quantum
Calogero model (1.19).
The outline of the thesis is as follows. In Chapter 2, we shall study two systematic formulations for the quantum Calogero model [76, 77, 80). One is the quantum La.x formulation,
which is a natural generalization of the classical Lax formulation deYeloped by the author and
his collaborators [3-l, 7~-78. 88, 89). The other is the Dunk! operator formulation whose fundamental properties were studied by Dunk! [27). This method was introduced by Polychronakos
[62) into the field of the in,·erse-square interaction models. Using these formulations, we shall
study a systematic construction of the mutually commuting conserved operators and the underlying symmetry of the quantum Calogero model. In Chapter 3, we shall present an algebraic
14
CHAPTER 1. INTRODUCTIO.V
·
·
f h
m Calogero Hami ltonian (1.19) and try a conconstruction of the etgenfunctwns o t e quantu
.
d
.
. [76 77 79) -\s
struction of the simultaneous eigenfunctions of commutmg conserve opetatOlsd
• ; . · ·d
we have mentioned in Section 1.1 , just presenting a set of commut mg consen·e opera ots an
constructin simul taneous eigen functions for them in practice are completely dtfferent pr oblems We s~all complete the a lgebraic construction of the energy etgenfunct tons [7~, ~7) ':'h1 c:1
.
·d d b . Perelomov [61) As a first step toward the rdenttficatwn o t 1e srmu was fi rst const ere
)
·
II d.
r th fi t
tan eo us eigenfunctions of the commuting conserved operators, we sha . tagona tze
e rs
two commu ting consen·ed operators and construct some s imul taneous etgen fun ctwns [79) .. In
Chapter 4 , we shall identify the simu lta neous eigenfuncttons of all the commutmg consen ed
t
[83 8-l 86 87) The simul taneous eigenfunctwns form t he orthogonal basts of the
:::~:1 °:~ is the ~as~ wi~h the orthogonal basis of the hydrogen atom. Using the Dunk! operator fo:mul ation, we shall prove the algebraic scheme, or m other \\"Ords, the Rodngues formul~
for the construction of the orth ogonal symmetric polynomials assoctated wtth .the orthogona
basis of the Calogero model, which we call the Hi-J ack (hidden-Jack) symmet n c polynomtals.
vVe shall st udy some properties of the Hi-Jack polynom ials and pro,·e that they mdeed form
the orthogonal basis of the Hilbert space of the quant um Caloge ro model. The fin a l chapter ts
devoted to summary and concluding remarks.
Chapter 2
Algebraic Struct ure of t he Calogero
Model
Construction of the conserved cu rrents and identifi cat ion of und erlying symmetry are important problems for integrable systems. For the classical integrable system, the Lax form ulation [47) has p layed a n imp ortant role to solve these problems. Howeve r for th e quantum
theory, the non-commutati vity of the canonical conjugate variables spoils some merits of the
Lax formul ation. And no general method to overcome this difficulty has been developed e,·en
by now.
For qu antum im·erse-square-interaction models such as the Calogero-:'l loser (1.20), Sutherland (1.21 ) and Calogero models (1. 19), two approaches were de,·eloped. One is the Dun k!
operato r formula tion [27), that "·as first applied to the above im·erse-square- interaction models
by Polychronakos [62). By the Dunk! operator form ulation, or in other words , the exc hange
operator formali sm, sets of commuting conse rved operators for the three models were obtained.
The other formul ation is the quantum La.x formula t ion [3-l, 74- 79,88, 89), which is a natural
quantum mechani cs generalizat ion of t he classical Lax formul ation. The quantum Lax formul ation was first introduced and st udied for the quantum Calogero-l\loser and Sutherland models
[34, 74, 75, 88, 89), and then extended to the Calogero model [76-78). The crucial points for th e
formulation are the "sum-to-zero" condition for the !\!-matrices and the recursive const ru ction
of the generalized Lax equations. Through the generalized Lax eq uat ions, we can compute the
commutators among the operators related to the three models. :\ set of commu t in g consen·ed
operators for the Ca logero-i'd oser model [74 , 75 , 88, 89) and the algebraic st ructures of the three
models [34, 75, 77, 78, 89) were obta ined through the quantum Lax formulation.
T he aim of this chapter is to present the quantum Lax and Dunk! operator formulation s,
which will play an important role throughout the thesis , and the !!"-symmetry structure of the
qua nt um Calogero model:
He
Pi
(2.1a)
=
a
-i-.
(2.1b)
8x1
~ loti vated by the classical Lax equatio n for the classical Calogero Hamiltonian, which has
been presented in C hapter 1, we formulate the quantum Lax equation for the quantum Calogero
15
16
CHAPTER 2. ALGEBRAIC STRUCTURE OF THE CALOGERO MODEL
2.1. QUANTUM LAX FORi'vfULATION
17
The sum-to-zero condition leads to a systematic construction of the conse rved operators, which
IS realized more easily by the trace formula in the classical case.
From the La.x equation (2.5), we see that the matrices of the following forms ,
Hamiltonian . Using th e quantum Lax equation for the Hamiltonian , we construct three families
of commuting operators, namely, commuting conserved operators and power-sum creatiOn and
annihilation operators by summing up all the elements of operator-valued matrices. By explicit
calculations, we get sume Lax-equation-like relations for some conserved operators and powersum creation and annihilation operators. These relations lead to the recursion relation for the
creation and annihilation operators. The recursion relation enables the recursive construction
of the Lax-equation-like relations which we call generalized Lax equat ions. The W-algebra
structure appears from the generalized La.x equations. Next, we summarize and reformulate
the Dunk! operator formulation for the Calogero model and consider its relationship with the
quantum La.x formulation. Integrability of the Calogero model is proved by explicitly show ing
the commuting consen·ed operators of the model.
satisfy a relation,
2.1
Using the sum-to-zero condition on the i\I-matrix, we find that the conserved operators are
obtamed by Simply summing up all the matrix elements of eq. (2.9):
Quantum Lax Formulation
In the classical theory, the Calogero model is solved by applying the projection method to
an N x N-Hermitian matrix whose time evolution is governed by the equation of motion for
the harmonic oscillator [57, 58]. The projection method also gives a way to introduce the Lax
equation for the classical Calogero model. Referring to the classical La.x equation (1.37), we
shall construct the La.x equation for the quantum model. Let us introduce three matrices:
1
ph+
ia(1- h ) -J 1
J
Xj -
Xk
(2.2)
,
(2.3)
ixjOjk ,
1
-a(1 - 1k) (
)
Xj - Xk 2
o
1
N
+ aOjk l=l
L
(
Xj -X!
)2 ·
(2.4)
(2.5)
where
(2.6)
and the constant a that appears in eqs. (2.2) and (2.4) is related to the coupling constant g by
g = a2
-
a.
(2.7)
The second term on the r.h.s. of the Lax equation (2.5) comes from the external harmon ic "·ell.
\\"e remark that the relation (2.7) contains a quantum correction. With h explicitly written,
the coupling constant g is expressed as g = a 2 - /ia, which reduces to the classical one (1.35)
in the classical limit h -; 0.
It is interesting to observe that t he matrix i\I is commo n to the quantum Calogero-i\loser
0) [3-!, 74, 75, 88, 89], and that the i\I-matrix satisfies the sum-to-zero
model (the case w
condition:
N
N
2:: ujk = 2:: ukj = o.
j=l
j=l
[(L+)P'(L -)"'' (L+)P'(L-)"'' · · ·,
(2.8)
u] + w(LPl- L
mk)(£+)P' (L -)"'' (L+)P'(L-)"'' ..
[(L+)P'( L-)"''(£+)P'(L-)"'' · .. , Mj.
(2.10)
:E
N
[He, L
~k=l
N
((L+)P'(L-)'"'(L+)P'(L-)"''. ·-)]
;k
[(L+)P' (£-)"'' (L+)P'(L-)"'' ... ,
1
0.
uL
(2.11)
Due to the sum-to-zero condition (2.8), the following identities hold for arbitrary operatorvalued matrix A:
N
N
N
L (Ai\I)jk = L Ajl(L Mk) = 0,
j,k=l
j,l=l
k=l
N
;,k=l
The Lax equation for the quantum Calogero model is expressed as
(2.9)
[He, (£+)P'(L-)""(£+)P'( L-)"'' .. ·]
L
l#j
L± = L±wQ,
(L+)P'(L-)"''(L+)P'(L-)"'' ... , PI + P2 + · · · = m1 + m2 · · · ,
N
(i\IA)jk
=
L
N
(2:: Mjl)Alk = 0.
l,k=l j=l
Then the last equal ity is readily verified by the above identities. Thus we can get various
expressiOns of conserved operators made of the same number of the L +- and L- -matrices
JUSt by changing the order of the two kinds of matrices. Among such conserved operators,
we are mterested in mutually commuting conserved operators, which can be simultaneously
diagonalized, and the conserved operators for which we can construct the "generalized" La.x
equatiOns recurs ively. Hereafter, we often deal with a sum of all the matrix elements. So we
N
denote a sum of all the clements of a matrix A byTE A
=L
.-l 1 k.
j k-1
A set of commuti ng consen·ed operators of the model i~ ~ne of the important objects for
quantum mtegrable systems because it shows the quantum integrability of the system in the
analogous way for the classical completely integrable systems. Since the most simple matrix
product of£+ and£- satisfying eq. (2.10) is£+ L-, we study the following consen·ed operators
as candidates of the commuting consen·ed operators:
ln=TE(L+L - )".
Explicit forms of the first two conseiTed operators are
(2.12)
CHAPTER 2 ALGEBRAIC STRUCTURE OF THE CALOGERO MODEL
18
2.1.
QUA.NTWVI LAX FORMULATION
+4(1- 6J k){-_!!_(p2
•
x2 _ J + PJPk
N
N1 1
N
LPJ + g L -y +w 2 L: x] + Nw(Na + (1- a))
j=t
J,k=t xJk
J=t
2Hc +constant,
TE(L+C) 2
N
~
1
J,k,l=t xJk xkl
1
N
-w 0 x·lxlk X (Pi+ PI+ Pk)- (aJ- 3a)-4 - a2 "'~{ __1__ - _1_}
l=t
J
;k
X
0l=t Xj1X1kXjk
2
2 ?
N
jk
XjiXlk
-a3
'\'I {
0
I= t
L
1
1 }
1 3 NI
1
_2_2_
+ _2_2_
+ -a
X;·IX;·k X1kX 1'k
2 l,m=l Xjl Xlk XjmXmk
}
J,k=t xJk
(2.16d)
N
W
4 '\'
0 x14
j=l
(2.13b)
1
where Xjk <jg x - xk· The symbol L means that all the indices in the summand must not
1
coincide. It is to be remarked that the total momentum is not a conserved operator. This
reflects the fact that the system (2.1) is not translationally invariant. Furthermore, we see that
the conserved operators generally possess the following form s:
N
In= LPJ" + ·'' ·
Note that in the explicit forms of the generalized Lax eq uation and the J\I-matrix for the
second
· and the \J. conserved operator , eqs. (?-· 16c) and (2 -16d) , th e genera 1·tzed Lax equat10n
;atnx for the first consen·ed operator appear. This is because th e second conserved opera-tor
· · of t 11e
12 contams
d the first conserved operator It ' as is seen in eq · (2 ·13b) - S uc h mtxmg
ower or er conserved operators generally occurs. As we see below the L + L-_ t · .
t ·
constant matrix terms,
,
ma nx con ams
(L + wQ)( L- wQ)
(£2- w2 Q2) + w[Q, L]
(2.14)
1
2(£+ L-
j;:::l
Thus we confirm that the first N conserved operators form a set of functionally independ ent
conserved operators of the model.
In order to confirm the integrability of the model , it is not enough just to construct a set of
independent conserved operators. We must verify the mutual commutativity of the conserved
operators (2.12):
(2.15 )
=0, n,m=1,2, .. ·.
[InJm]
We expect that non-tri,·ial conserved operators also satisfy the La:x-equation-like relations with
J\I-matrices that meet the sum-to-zero condition, as was demonstrated for the case w = 0 [75].
We generally call such Lax-equation-like relations generalized Lax equations. Using the explicit
forms of the conserved operators (2.13), we obtain the generalized Lax equations for It and ! 2 :
[h.L±]
xJk
1
+4w(a- 1- Na)Hc + w2{N(a- 1- Naf- aN(N- 1)} ,
Mt
. 1 (Pi - pk)
1a3
L ~~+(g 2 -g) L ~
N
N
2
2
N
2+ 2
1
xi +xk
2 '\' 1 xj.- xk}
+w2{"'(
x12p12 + p12x12) + g '\'
- +
2 -a 0
0
0
j=:l
j,k=l
Xjk
j,k,m=l X;mXmk
[lt,L±]
+ Pk2) -
1
. 2 '\'I
(2.13a)
N
N1{
1
1
1
1 2 1
1
1
1 2}
2 1 + PJ-PkLP~ + g L
Pj2
+ PJTPJ + -pk~ + ~Pk~Pj + 2Pj
j=t
j,k=t
Xjk
Xjk Xjk
Xjk
Xjk X;k
X;k X;k
Xjk
+2l
;k
N
19
[1± , Mt] ± 2wL±,
(2.16a)
2.\J,
(2.16b)
[1±, J\h] ± 4wL±£'f£± ± 4w2 (a -1- Na)L±,
(2.16c)
+ L- £+) + w((a-
1)1- aT),
(2 .17)
where T is a matrix whose elements are all equal to 1:
(2.18)
We note that a sum of all the elements of the matrix product AtT A 2 for any operator-valued
matnces At and A2 is the product of the sums of all the elements of the two matrices:
(2.19)
Thus a sum of all the elements of a power of L + L- -matrix become a polynomial of th e sums
of all the elements of lower-order powers of the matrix
~ (L + L- + L- L +):
TE(£+ L-)"
1
T1:(2 (£+ L- + L- L+) + w((a- 1) 1- aT))"
TE(~ (£+ 1- + L- £ +))"
+
~ T1:(~(L+ L-
+ L- £+) r-tw((a- 1)1-
T1: (~(£+ L- + L- £ +)
"
1
r
aT)(~(£+ L- + L- £+) )"-' +
+ nw(a- 1) TE(~ (L+ L- + L- L+) )"-t
-wa LTE( - (£+ L- + L- £+))'-t TE(~(L+ L- + L- £+))"-' + ...
2
=t
2
20
2.1. QUANTUM LAX FORMULATION
The constant matrix terms generate a polynomial of lower order conserved operators in this
way. That is why the conserved operators I .. (2.12) contain lower order conserved operators.
1
2 2
2
Instead of the L + L- -matrix, we adopt the matrix {L + L- + L- L +} = L - w Q and remove
+4(1- hl{-_::_(p2
+
J
2
i
+ Pk2)
1
. 2 '\''
2
-ta L
X·IX
1=1
N
the constant matrix terms. Then we obtain a modified formula for conserved operators:
PiPk
X 1k
N
21
J
-
.
1
t a - ( p1 -
x\J
Pk)
1
X · (pj +PI+ Pk)lk ;k
N
(a3- 3a)-4 - a2 '\''{ __1__ x jk
L1=1 XjiXIkXjk
2
_1_}
2 2
XjiXIk
-aJI:'{~+~}+~aJ
(2.20)
1=1
The first two modified consen·ed operators are given by
+4w
{'
1
}
2 l,m= 1 Xj1X1kXjmXmk
'', x 2 - x ·x + x 2
2
2
J
J2 I
I - a(1- .Sjk) xi - XjXk + xk }.
1=1
xil
xJk
XjiXjk
2{ a.Sjk
XlkXjk
L
(2.22d)
This formula guarantees that the conserved operators J.. (2.20) and hence In mutuallv commute. Howe\·er, recurs1ve constructwn of the generalized Lax equations for these COI;sen·ed
o~crators has not been successful because of the difficulty of the order problem of the L + _ and
L -mat~1ces. Equation (2.22) for the first two cases n = 1 2 are confirmed by the e'"pli·c·t f .
of the i\I m t .·
(?
"'
I 01 ms
)
'
.. - a uces, e[qs. _ -]· 22c and (2 .22d). This pro,·es the mutual commutativity of the
conserved operators, I .. , Im = _o and equivalently
= 0 for n = 1, and m = 1, 2, . . ·.
But now, the cases n :0:: 3 remam to be a conjecture. The mutual commutativity of the conserved operators for genenral cases will be verified by use of the Dunkl operator formulat·o ·
Section 2.4.
I n Ill
[I.. ,Im]
Namely, i,, corresponds to the highest order conserved operator in I ... Note that the coefficient
of the fourth term of the second conserved operator, g2 - g, contains the quantum correction.
With the Planck constant h explicity written, the coefficient is expressed as l- hg. Moreover ,
the coefficient of the third term in the w2-order terms is not expressed as a polynomial of
the coupling constant g, but as a2 These facts show the quantum conserved operators of the
Calogero model are not given by the translations of the classical conserved quantities by the
correspondence principle [20, 59]. We expect that the generalized Lax equations can be obtained
for the higher order conserved operators. From eq. (2.16), the generalized Lax equation for I,
" For rdecurdsive construction of the generalized Lax equations, it is convenient to introduce
\ ·vey 1 or ere product:
(2.23)
In part icular, the matrix,
is to be expressed as
[i.. ,L±j =
[L±,J~I,j ±2nw(L±L'~')"- 1 L±,
N
[(L+)"(L-)''lw'
(2.22a)
(2.22b)
[H, [(t+)"(L-)"Jwj
j=1
where explicit forms of the first two
.\In
matrices are
(2.24)
satisfies a relation that is similar to eq. (2.10):
N
'fJAin)Jk = 0, 'fJAI.. )ki = 0,
= [[(L+)"(LTJw ,
Mj ,
(2.25)
Thus we obtain a formula for a set of consen·ed operators:
0~
(1\It)jk
(i\-I2 )ik
2
= Td(L+)"(L-)"j,v
(2.26)
~h;se conserved operators do not generally commute among themselves. Howc,·er, the choice
(---6) IS convement for the mvestigation of the algebraic structure of the model.
Explicit forms of the first two consen·ed operators are
o:
=
TE~(L+L-+L-£+)
N2
/\·,1
L Pj + 9 L
;=I
-y
j,k=1 xJk
?N
+ w- I: x12
i=I
CHAPTER 2. ALGEBR.J.IC STRUCTURE OF THE CALOGERO MODEL
22
(2.27a)
23
where p and m arc non-negati\'e integers. In the remaining part of this section, we shall
recursively construct the generalized Lax equations for this family of operators.
First, we introduce two series of operators that will play an important role in an algebraic
treatment of the eigenfunctions of the model. Let us consider operators defined by
o;<jg'EJ
TE(L+t,
(2.30a)
0~~ En
TE(L-t,
(2.30b)
where n is a nonnegati\'e integer. By use of the Lax equation (2.5) , commutation relations
between the Hamiltonian (3.2a) and these operators are calculated as
[Hc,TE(L+tj
TE{[(t+t,M]
(2.27b)
+nw(L+t}
nwE1,
Using the explicit forms of the conserved operators
and 0~:
equations for
ot
[ot, L±]
i\!11
[o~,L±]
[L±, ut'] ± 2wL±,
2J\I,
(2.28a)
TE{[(L-t,AI]-nw(LT}
(2.28b)
-nwEn.
(2.28c)
[L±,ui] ±4w[(L±) L"'Jw,
2
N{a-dP2 + PJPq + Pq)2 + w3(PJ.
1
)
4h L
Pq
J q=I Xjq
q¥J
J
Xjq
N
1
1
l=l
Xj1XIqXjq
;q
1
1 } 1 3 ;,
+a3"' - - - - --a L
2
1
N, {
X~;t·X~;q
x1q X~JQ
a 2
+4(1- OJk) { --::["(Pi+ PJPk
xik
N,
2
N '{
1
1 }
x x x2 + x 2 xl
J1 1q Jq
J1 q
}
I ,m-I XjtXtqXjmXmq
L
-a3
1
1
1 3
LN' {--+-?-?-}+-a
.;_,
L
1
1
The commutators between E1, n = 1, 2, 3, and £± are
[E[,L±j
[E1,£±j
2
_ ____:___ _
[L±,MJj-w(1'f1)1,
(2.32a)
0,
(2.32b)
[1±, AI~]- 2w(1 'f 1)£+,
1
x;k
3
Similar to the creation and annihilation operators in the theory of quantum harmonic oscillator,
(MJ)Jk
N'{--1__ - _1_}
---(PJ +PI+ Pk) - (a - 3a):il- a
x 1XIkx2x21Xfk
Xj1XIkXjk
;k
1=1
J
;k
J
-ia2
l=l
1=1
. 1 (Pi- Pk )
+ Pk2) - Ia3
1
(2 .31b)
these operators change energy eigenvalue by nw. \Ve shall call the operators E1 and En po"·ersum creation and annihilation operators, respectively. The reason why we add power-sum to
the name will be clear in Section 2.4 and Chapter 4.
2
3
+ia 2 2:.:' ---(Pi+ PI+ Pq) + (a - 3a) x• +a L
~
/ :;:1
(2.31a)
[Hc,TE(LT]
L
}
l=l x] 1x]k
XfkX]k
2 l,m;::;l XjtXtkXjmXmk
2
4 2{
N x1 + XjXI + Xf
XJ + XjXk +
+-w ac51k L
_2
- a(1 - OJk)
.2
3
I=I
x 11
x 1k
Xk} .
-2a(1 - o1k) 2
xik
(2.32c)
N
1
I=I
xil
+ 2ao1k 2: 2
1¥j
,
(2.32d)
(2.32e)
(2.28d)
(2.32f)
1¥1
As we have expected, the above J\I-matrices (2.28b) and (2.28d) satisfy the sum-to-zero cond ition. The first relation (2.28a) is nothing but the Lax equation for the Hamiltonian. We expect
that similar relations also hold for all the operators,
We remark here that three M-matrices shown abO\·e meet the sum-to-zero condition. In the
same way, we can compute the commutators between En, n = 1, 2, 3, and £±:
(2.29)
(2.33a)
24
(J\f~)jk
[B
2,
L±]
0,
(2.33b)
[£ ±, M~]
+ 2w(1 ±
1
-2a(1- 1k) 2
o
[£± , M~]
1)£-,
(2.33c)
N
1
1=1
l;<j
xil
+ 2aOjk I: 2•
xik
t
(2.33e)
1
1
2
+ Pm) -
Xjm
m=l
N
1
I:--.
.
I
Xj
n
.
Xj+Xm}
3twa-2-
3
4
~(£+) 1 - 1 { (£+) 2£- + £+ L-£+ + £-(£+n(£+)"-']
n
Tr { [3w ~ { (£+)'-!{ (£+)2 L-
+ Xk
-8nw 2 B!+I.
Xjm
These relations show that commutators [EX, Bl] and [En, BmJ, n = 1, 2, 3, m = 1, 2, ·· · vanish.
The explicit forms of generalized Lax equations for the power-sum creation and annihilation
operators, eqs. (2.32) and (2.33), lead us to an expectation that the commutation relations
(2.38)
Here we have used again the sum-to-zero condition on the MJ-matrix. This is nothing but the
recursion formula (2.36).
Preparations for the proofs of eqs. (2 .34a) and (2.35a) are finished . We have alread,·
obtained explicit expressions of the first three, eq. (2.32). By the inducti,·e assumption, ":e
assume that the relatiOns (2.34a) hold upton= p with i\!6-matrices satisfying the sum-to-zero
condition. We want to calculate a commutator [BJ+I• £±]. Using eq. (2.36), we ha,·e
= 1, 2, 3, · · ·, take the following forms,
[£±, M;]- nw(1 'f 1)(£+)"- 1,
[£±, u;j + nw(1 ± 1)(£ -)"- 1,
(2.34a)
(2.3.Jb)
(2.39)
By use of Jacobi 's identity for the commutator,
with the AI-matrices satisfying the sum-to-zero condition:
N
L(i\I;)Jk
+ £+ L- £+ + L-(L+n(£+)"-', M~]- 8nw2(£+)"+1}
(2.33f)
.
m;<j
[EX,£±] and [En , £±], n
4
[Bt, wTr
-(1-o1k){3a 2 (p1 +pk)+3ia
--3twa-.2- }
Xjk
l=l XjiXikXjk
Xjk
~{
\Ve shouldfrema(r k he)re that we have used the sum-to-zero condition of the .\!?-matrix. SubstitutiOn o eq. 2.3 7 mto the l.h.s. of eq. (2.36) yields
(2.33d)
+ 3w(1 ± 1)(£-j2,
+Ojk L., 3a-2-(Pi
[A, [B, C]J + [B, [C, A]]+ [C, [.4., B]J
N
(2.35a)
L:(Mo"hi = 0,
= 0,
(2.40)
the r.h.s. of eq. (2.39) is rewritten as
j=l
N
N
L(M~)Jk
L(AI~)ki
j=l
j=l
= 0.
(2.41)
(2.35b)
In order to proceed the calculation, we need a formula:
As a first step toward the recursive construction of the generalized Lax equations for all the
operators Ofn , we shall prove the relations (2.34) and (2.35). The proof is done by induction.
To do an inducti,·e proof, we need a recursion formula for the power-sum creation operators:
(2 .36)
[[o~,BJj,L±]
[£± ,
[MJ,M6] + [Mi, B}] + [o~ , M6]]
4
{
p-1
-3w2p (1 'f 1) L(L+)l-1 { (£+)2 L-
+ £+ L- £+ + L -(L +)2}(£ +)p-l-1
1= 1
The proof of eq. (2.36) is carried out as follows . Commutator between
0~
and
BA is
+ (1
'f
l){ 2(£+)P L- + 2L -(£ +)P + £+ L -(L +)p-I + (L +)p-I L- £+}
- 3(1 ± !J(£+rl }·
(2..J2)
This formula is pro,·cd by repeated use of Jacobi's identity (2.40) and the Lax equations, (2.28c)
and (2.34a) for n = p. Applying Jacobi 's identity to the l.h.s. of eq. (2 ..J2) , we haw
(2..J3)
CHAPTER 2
26
ALGEBRAIC STRUCTURE OF THE CALOGERO A-IODEL
. use of the Lax equatiOns,
·
(2 ·28 c ) an d (2.34a) for n = p, and applying Jacobi's identity
l\lakmg
again, \\·e ha,·e
-[[Bj,L±],Oi]- [[L±,oi],Bj]
=
2.1. QUANTUM LAX FOR.\IULATION
After a calculation which is similar to that in eq. (2.44) , we obtain
Tools for the calculation are Jacobi's identity and eqs.
Mg+ 1-matrix as
1
-[[L± , ugj-pw(1=t=1)(U)P- ,0i]
(2.32a) and (2.42).
1 [ G] .
1\[,P+I
o - 8pw2 BIt '
+ [[L±, MJ] ± ~w{(L±)2 L '~' + L± L'f L± + U(£±)2}, Bj]
~w[Bj, (L±? L'f + L± L'~' L± + £'f(L±) 2]
(2.~8)
[uJ,Bt] + [oi,M6]j + [[L±,MJj,M6] + [[M6,L±j,Mij
_ [oi, pw( 1 'f 1)(L +r 1] + [pw(1 'f l)(L+)p-I , Mi]
which is exactly the same as the formula (2.34a) for n = p+ 1. As we have confirmed before, the
formulae, (2.34a) and (2.35a), hold for n = 1, 2, 3. Thus we complete the proof of the formula
for a ll positive integer n.
In a sim ilar fashion, we can deduce the relations for the annihilation operators, eqs. (2.3~b)
and (2.35b). Instead, we can easily check them from the generalized Lax equations for the
creation operators (2.3~a) and (2 .35a), which has been formulated just now. Computing the
Hermitian conjugate of the relation (2.34a) , we have
'f~w[Bj, (L±) 2 L'~' + L± £'f L± + L'f(L±) 2]
±~w[(L±) 2 L'~' + L±L'~'L± + L'~'(L±) 2 , M6]
[L±, [MJ, M6] + [MJ, Bj] + [oi, M6]]
-~w 2 p{(l 'f 1) ~(r+)'- 1 {(L+? L- + L+ L- L + + L -(L +) 2}(L +)p-I-I
(2.~9)
1=1
Defining the i\J~-matrix as
+ (1 'f 1){ 2(L +)P L- + 2L-(L +)P + L+ L -(L+y-I + (L +y-I L- r+}
( 2 .4~)
- 3(1 ± l)(L+)p+I }·
Consequently, we obtain the formula (2.42).
following symbols:
[oi,BJJ,
G
[Mi, ug] + [Mi, Bj] + [oi , ug].
I] + [[L ±, Md], K] - w( 1 'f 1) [1, K]
(2.50)
N
IJM,~)jk
IJ i\I~)kj = 0.
j=l
j=l
(2.51)
The commuting power-sum creation operators will play an important role in the algebraic
construction of the eigenfunctions of the Calogero Hamiltonian (2.1) in Chapter 3.
The next task is a recursi,·e construction of the generalized Lax equations for the operators
0~, defined by cq. (2.29). \\'atching closely on the relations (2.28a), (2.28c) and (2.3~), we
notice that the following relations might hold for 0~,:
p-1
[L±,
+~ [w 2 p{ (1 'f 1) 2_]L +) 1- 1 { (L+) 2 L- + L + L- L + + L-(L +?}(L +)p-I - I
M;,,] + mw(1 ± l)[(L+)P(L-)'n-tv
-pw(1 'f ll[(£+)"-I(L-)'tv'
1=1
+ (1 'f 1){ 2(L+)PL- + 2L -(L+)P + L+ L -(L+)p-I + (L+)P- 1 L- L+}
- 3(1 ± 1)(£+)P+l }, B/].
(M~)t,
We remark here again that eqs. (2.3-la) - (2.35b) assure the commutativity among the creation
operators and among the annihilation operators:
-[[K,L±],Bi]- [[L±,B/J,K]
3
=
[L±, M~j + nw(1 ± 1)(L -)"- 1 ,
N
Equation (2.42) is nothing but the generalized La.x equation for the abo,·e ]( operator. Note
that the operator-valued matrix G also sat isfies the sum-to-zero cond itiOn because the matnccs
MJ and M6 do.
-With eqs. (2.32a), (2.32b) and (2.42), the r.h.s. of eq. (2.41) is calculated as follows:
- [ [ L ±, G], B
M~
we get the generalized Lax equation for the annihilation operator Bn (2.3-lb):
For brevity of the expressions, we adopt the
](
(2.4 7)
We remark that the Mg+ -matrix satisfies the sum-to-zero condition, for the matrix G satisfies
it. Then from eqs. (2.39) , (2.46) and (2.47), we finally obtain the expected expression ,
[L±,
3
We define the
1
[L±, [uJ , Bj] + [oi,M6]] + [[Oi,L±j,M6]- [[Bj,L±j,MJ]
_ [Oi,pw(1 'f 1)(L+r 1] 'f
27
N
(2.~5)
(2.52a)
N
IJ.\I;:.)jk
IJM;,.)kj = 0.
j=l
j=l
(2.52b)
2.2.
28
W-SYAI.\fETRY OF THE CALOGERO MODEL
which meets the sum-to-zero cond iti on. Finally we obtai th
. t d f
1 f
(2. 6) and ( . ):
'
n e expec e ormu a rom eqs.
2 57
5
Note that the relations hold for m = 0 (2.34a), p = 0 (2.34b) and p = m = n for n = 1, 2
(2.28a) and (2.28c). We shall prove eq. (2.52) by induction.
First, we prove a recursion formula for o;~-m:
[L± ' ~vr-~-~]
1'+1
(2 .53)
n-m] == 4w (n- m )O"-mm+l 1.
[B2, Om
29
+(p + 1)w(1 ± 1) [(L +)"-~- ~ (L -vJw
This formula can be verified as follows. Substitution of the definition of o;:,-m yields
-(n- 1-' -1)w(1 'f 1)[(£+)•-~-2(£-)~ + t]w·
(2.59)
(2.54)
Using eq. (2.33c) and the sum-to-zero cond ition on the Mg-matrix (2.33d), we get the recursion
Thus the pwof is completed. The generalized Lax equations provide an easy way to consider
the algebratc structure of the Calogero model.
formula:
[B 2,o;:,-m]
=
=
1
Tx:{[[(L+)"-m(L-)mJw, Mgj + 4(n- m)w[(L+)"-m- (£-)m+tJ
4w(n- m)0;:,+';'-
1
2.2
(2.55)
Now we shall prove eq. (2.52a). In the case of m = 0, the relation holds. We assume that
the relations (2.52a) hold up to m = !', and that the M;;,-m-matrices satisfy the sum-to-zero
condition. Due to the formula (2.53), the commutator [o;:;:r-t, £±] is expressed as
(2 .56)
W-Symmetry of the Calogero Model
. The generalized Lax equat ions reveal an interesting algebra among the operators OP . Let
us mtroduce a new operator by
m
w<•l
n
[11",\'l, L±j
=
-[[o;-~,L±],B2]- [[L±,B2],o;-~]
_
N
+ [(n- p)w(1 'f 1) [(L +)"-~- (L -)~Jw, B2]
1
[L±, [82, J\I;-~j + [M~, o:-~]j
± 1) [[ (L+)"-~(L -)~-tv' Mg]
Jw
-4(n- 1-'- 1) (n- !L)w2(1 'f 1) [(L +)"- 1' - 2(£
'fJM~'l)Jk
'fJM~'l)kJ = 0.
j=1
j=1
..tw
-)"+llw
(2.61b)
(2.61c)
IV
w'
u<•l]
n
1
+ (s + n- 1)(t _ m _ 1l [(L+J•+t-n-m-3(£ -J•+t+•+m-3Jw
2
- 1 (s 2
± 1)[(L+)"- 1' - 1(L-)"Jw
-4(n- !'- 1)(n- !L)w2(1 'f 1) [(L +)"-~- 2 ( £ -)~+ 1 ]
.
1
w(n - !') { [82 , u;-~] + [ug, o:-~] + [ug, M;;-'']} ,
4
n - 1)(t + m - 1) [(L +)s+t-n-m-J(L -)'+t+n+m-Jlw + ... }
+ p n(s,t)(IV("))
( n(t -1)- m(s -1J)lr<•+t-2)
n+m
,m
l
'
(2.57)
(2.62)
p(s,t)(ll'("))
· an unspect·fi ed polynomial of 11' (u) -operators, u :::; s + t - 3, l :::; n + m,
where
, .
n,m
1
ts
1
htch are generated by replacements of L + and L-,
We define the JII~'.;:i- 1 -matrix by
M~'.;:t'- 1 =
(2.61a)
N
TE{_.!._ [[(L +Jt-m-1 (L-)'+m-1]
M;:-''] + [u~, o;;-~] + [ug, u,~-~]]
+4(1' + 1)(n -l')w 2(1
w'
[11',\'l, T E [(£+)t-m-1(L -)t+m-tvl
-(n- !L)w(1 'f 1) [[(£+)"- 1' - 1(£ -)'tv' ug] - 2w(1 ± 1*-, i\I~'- 1 ']
[L±, [82,
4
_.!._ 1\[s-n-1
4W · s+rt-l'
From the abo,·e generalized Lax equations for the w <•l_operato r the commutation relation
"
'
among the n ·~·l-operators is calculated as
+[[82,L±j,M;-'']- [[o;-'',L±j,Mgj
± 1) [(L +)"-~- 1 (L -)~
(2.60)
[L±, i\I~'l]
1
+4(s + n- 1)(1 ± 1) [(L+)'-n-1 (L -)s+n-2L
--(s- n -1)(1 'f 1)[(L+)s-n-2(£-)Hn-1]
± 1)£-, o;-'']
± 1)[(£+)"-~(L-)~-tv' B2]
+4(p + 2)(n- p)w 2(1
s >_ ]71] + 1.
1
- [[1±, M;-~] , B2] + [[1± , Mgj, o;-~j + [2w(1
+!Lw(1
4
In this notation , the genera liz ed Lax equations (2.52) are written as
By iterated use of Jacobi's identity (2.40) and the generalized Lax equat ions for B2 (2.33c) and
o:-~ (2.52a), the r.h.s . of eq. (2.56) is calculated as follows:
-[1Lw(1
= _.!._os-n-1
W
s+n-1,
-
11
(2.58)
(2.63)
30
=
Here 1 is a unit matrix and Tis a constant matrix whose matrix elements are all 1: Tjk
l.
This,is the IV-algebra in the sense the algebra is a higher-spin generalization of the Virasoro
algebra (for review, see ref. [16]). We remark that indices sand n are integer or half-odd integer,
though of course both s-n-1 and s+n-1 must be non-negati1·e integer. The algebra possesses
the operators with indices s = l, 3/2, 2, 5/2, · · ·, which represent the conformal spins.
For the classical mechanics case, derivation of the IV -symmetry was examined by two different methods: the collecti1·e field theory [3] and the classical r-matrix method [7]. The classical
r-matrix approach 11·as extended to the trigonometric and elliptic versions of the Calogeroi\loser model [6, 8, 66]. Quantum generalization of the collective field theory [4, 5] also suggests
the IV-symmetry structure of the quantum Calogero model, though the relationship between
the quantum Calogero model and the quantum collective field theory has not been estab lished.
Our proof is the first direct proof of the 11'-symmetry of the quantum Calogero model. For
the spin generalizations of the quantum Calogero-i\loser, Sutherland and Calogero models were
also investigated by the quantum Lax formulation [34, 78]. The sum-to-zero condition was also
an important key in the investigations.
At the end of this section, we gi1·e a brief summary of the correspondence of the lV~')
operators to the creation operators, the annihilation operators and the consen·ed operators:
2.3. DUNKL OPERATOR FOR.\IULA.TIO.V
31
=
where a complex phase factor J.L exp(i1rB) corresponds to the statistics of the particles in the
system 11·hose meaning will be described later. The coordinate exchange operators J(lk obey
J(lk = J(kl, (Ktk)
KtkAt
= Akf(lk.
2
,i, KL
=
= fJ.- 2J(Ik,
= A 1K 1k,
KtkAJ
for j
#
l, k,
11·here AJ is either a momentum operator p1 , a particle coordinates x 1 , or particle permutation
operators Kj; , i = 1, 2, · · ·, N. These relations (2.65) guarantee that the coupled momentum
operators (2.64) are Hermitian operators, 1rJ = 1r1. By linear combinations of the coupled
momentum operators and the coordinate operators ,
(2.66)
we define creation-like and annihilation-like operators :
(2.67a)
(2.67b)
Ct
c)
The creation-annihilation-like operators, {c~, c)ll = l , 2, · · · , N}, describe the algebraic structure of the Calogero model. On the other hand , the coupled-momentum operators and the
coordinate operators, {7rt, qdl = l , 2, · · ·, N}, are applied to the study on the algebraic structure of the Calogero-i\loser model and the Sutherland model, which will be summarized in
Chapter 4.
l. creation operator
2. annihilation operator
3. conserved operator
In particular, the creation operators and the annihilation operators respectively form commuting subalgebras of eq. (2.62). The property of the creation operators and the annihilation
operators will play an important role in the algebraic construction of the energy eigenfunctions
in Chapter 3.
By a straightforward calculation, we can check the following commutation relations for the
creation- and annihilation-like operators:
[c1,
C.n] = 0,
(2.68a)
[c/, c~] = 0,
2.3
Dunkl Operator Formulation
[c1, c~] =
In the researches on the algebraic str uctures of the Calogero model, two formulation s
were independently developed. In the last two sections, we have formulated the quantum Lax
formulation [34, 74-77] that is an extension of the Lax formulation in the classical theory [52].
On the other hand , Polychronakos introduced another formulation for these models [62] that he
called exchange operator formalism. His method is based on the differential operators including
coordinate exchange operators. Now Polychronakos ' method is usually called Dunk! operator
formulation because the basic properties of this kind of differential-difference operators were
first studied by Dunk! [27]. These two approaches independe ntly reveal interesting aspects of
the models. \\·e shall study the relationship between the two, which have not been checked by
serious considerations.
(2.68b)
+ a{t- 1L
N
2wc5tm(l
.
Ia{!
-l
LN
1
k=1 Xi- Xk
k;tl
/
-.--!Ilk,
l
= 1, 2, . .. , N,
(2.6~)
1
Ktk)- 2waf.t- (l- c5tm)Ktm·
(2.68c)
ktl
k=1
Using eq. (2.68), we can construct a set of commuting operators that correspond to the
conserved operators of the Calogero model:
N
1,.
=L(c/cl)",
(2.69a)
1=1
[1,., l,n] = 0,
n,m = 1, 2, ·
(2.69b)
This relation is ,·erified as follows. \Ve introduce the number-like operators as
We summarize the Dunk! operator formulation d eve loped by Polychronakos [62]. We introduce differential operators which he called coupled momentum operators,
1rt =PI+
(2.65)
(2.70)
Commutators among the number-like operators are gi1·en by
(2.71)
-
--
-----
32
-
Using the above commutation relation, we can compute the commutator among the operators
2.4. RELATIO.VSHIP BETWEEN THE TWO FORlv!ULATIONS
33
we notice the following relations for arbitrary polynomials P:
(2.69a) as follows:
(2.73)
N
L
j,k=l
N
[(n1 )", (nk)m]
n
They can be proYed by induction on the degree of a polynomial. First, we consider the simplest
case, namely the hnear case. There are only two independent elements and we can check the
'
relation (2. 73) directly:
m
L L :L(nj)P- 1(nk)q-l [nj, nk] (nj)"-P(nk)m-q
j,k=l p=l q=l
N
n
m
L L :L(n1)P- 1(nk)q- 12waJ.L- 1(nk- nj)Kjk(nk)m-q(n1 )"-P
.
(PI+ 1wx1
j,k=l p=l q=l
N
2waJ.L-l
:L(nj)P- 1((nk)m- (nj)m)Kjk(nj)"-p
-
n
n-1
n+m-1
r=m
r=O
m-1
L -L
r=O
+
L
k=l
(
L
L(kl ,
(27-!a)
~
.
PI - lWX1
. -l
+ lJ.L
a L~ - -1- K1k) I
k=l XI- Xk
-)l
N
1
:L(61k(PI- iwxl) + ia(1- 61k)-
r=n
k=I
\lt(Xt, · · · , Xk, · · ·, Xt,''', XN)
where the phase factor J.L giYes the informat ion on the statistics of the part icles. For instance,
the case I' = 1 corresponds to the boson system and the case I·' = -1 corresponds to the fermion
system. In order to include the fractional statistics [90], we allow 1-' to be a complex number,
1-' = exp(i1rll). Existence of such wa,·e functions is known for the Calogero mode l [19, 76, 77].
~~
(2.74b)
~
For all the independent elements of the polynomials up to degree d, we assume that the relation
(2.7() holds. Allwe haYe to do is to prove the relation for c/P(cf,cl) and c1P(cf,c1), where
P(c1, cl) IS an arbitrary polynomial of degree d. It can be done by a simple and straightforward
calcu lation:
f_ ~I<Ik)P(c/, c1)1
the restriction of the operand to such waw functions. Then
Xk
N
:L(61dP1
k=l
N
N
~
1
+ iwxl) + ia(1- 61k)-.- -)P(a!, ak)l
X1- Xk
~
1
{;; L(k ,~ P(L+, L-)k{
N
(2.72)
Jt
f_ Likl .
k=l
k=! Xt
k,<l
In the quantum theory, identical particles are indistinguishable. This requires the wa,·c
functions to be invariant up to a phase factor under the permutations of particle indices.
The action of the coordinate exchange operators to such wa,·e functions w(x 1 , x 2 , · · ·, x,v) is
described as
· · ·, Xk, · · · , XN ),
Xt- Xk
(PI+ iwx1 + iJ.L- 1a
R e lationship b etween the Two Formulations
,u\lt(x 1 , · · ·, Xt,
~
k,<l
(nj)' Kjk(nj)"+m-,-1
Thus we have confirmed eq. (2.69). 1 ote that the abm·e commuting operators contain coordinate exchange operators. If we restrict the operand to the states of identical and indistinguishable particles, we can obtain the differential operators that do not include particle permutation
operators. These differential operators generated by the restriction must be related to the
operators in the quantum Lax formu lation . We shall reveal the relationship in the following
section.
From now on, we shall denote by
1
n+m-1)
0.
2.4
+ iwxl) + ia(1- 61k) x _ x) ~~
N
~(n1 )q-l((nk)"- (nj)")Kjk(nj)m-q}
waJ.L-1 (:L-
Jl
1
N
{;; (61k(PI
waJ.L-1 j~l :;;(njy-l((nk)m- (nj)m)Kjk(nj)"-P
{
1
---]{
1k)l
k,<l
n
L
N
k=l Xt- Xk
j,k=! p=l
N
.
+ lJ.L- 1a L
{;; (L+ P(L+, L-) ) 1kl,,'
(PI- iwx1
+ iJ.L- 1a f_
(2.75a)
--1--_Kik)P(c/, cl)l
k=!Xt-Xk
k,<l
Jt
N
1 )P(cl,ck)
:L(61dP1iwx1) + ia(1- 61k)-.--.
k=I
Xt -
Xk
I
Jl
34
N
L
k=l
,,.
Lik
L
P(L+, L-)km
m=l
I
Jl
k=l
(2.75b)
f P2(c~, Cm)~~
1
= Lt TE P2(L+, L
-)~~ +
I.J
£ P2 (c~, em)JI
m=l
c)
=
E
Lit TE F1kP2(L+, L- )EI{·
(2.82)
k;i/
Using an identity,
Thus we have proved the relation (2. 73) for arbitrary polynomials.
i\loreo1·er, we can also see the correspondence of the commutator algebras,
l;;;l
35
where P 2 is an arbitrary polynomial. Thus we obtain
f(L-P(L+,L-)) 1kl·
[i:P1 (cJ,c1),
2.4. RELATIONSHIP BETWEEN THE T\VO FORAIULATIONS
[TEPI(L+, £-),TEP2(L+,£-)jl~'
··· ,If
··· ,If
[1, 1, · · ·, 1)EikP2(L +, L -)Eik[1, 1,
[1 , 1, · · ·, 1)P2(£+, L-)[1 , 1,
TE P2(£+, L-),
(2.76)
(2.83)
J.J
where p 1 and p 2 are arbitrary polynomials. To prove the above relations, it is sufficient to
show the following identities:
where a superscript
T
means a transposition, we get
N
c)
N
f/2(c~,em)l~ = {;LitTEP2(£+,£-)I~·
(2.8-l)
(2.77)
This shows the validity of eq. (2. 77) for the simplest case. Following the same logic, we can
also confirm another simple case:
In a similar way to the proof of eq. (2.73), we shall prove the identity (2.77) by induction on
the degree of P1 • First, we get the following expression by applying eq. (2.73) on P2:
(2.78)
We shall consider a simple case, namely P 1 (c)) =c). From the explicit form of the£± matrix,
we can see the action of K 1k operator on £± as
(2.79)
N
c1 f
N
P2(c~, em) I~={; Lik TE P2(L+, £-)~~·
1
(2.85)
We assume that the relation (2. 77) holds for all the polynomials P 1 up to degree d. For a
polynomial of degree d + 1, cj P 1 (c) , c1), we haw
N
c) P1 (cJ, Ct)
1
.~ P2(c~, em)~~
Lt P1 (cJ, ct) TE P2(£+, L -)~~
=
N
+I.: L(kP1(c!, ck) TE FtkPz(L+, L -)Eikl
where an N x N numerical matrix F1k is given by
k=l
J.J
k;i/
N
k
L
L(kP1 (ck, ck) TE P2(£+, L
k=l
N
{;{£+PI(£+, L-)
-)1
I'
L TE P2(£+, L-)L.
(2.86)
0
In a similar way, we can also confirm
(2.80)
(2.87)
k
0
Thus we have inducti1·ely pro1·ed the correspondence (2.77). These relations (2.73) and (2.76)
assure that t11·o series of operators,
N
6:;, =I.: [(clJ"(ck)"'Jw,
1\ote that (Eik) 2 is the identity matrix. So the action (2.79) lead us to
(2.88a)
k~l
(2.81)
l\r(•) =
n
-
_2_6•-n-1
s+n-1'
4W
(2.88b)
36
respectively correspond to the operators Oi:, (2 .29) and
ll',\'>
(2 .60) and also satisfy the com-
mutation relation of the quantum IF-algebra under the restriction
~~.
2.4. RELA.TIO.VSHIP BET\\ BEN THE T\\'0 FOR.\IULATIO.VS
Summing up the above two terms , we have a commutator:
As a special case, we
have
37
[(cj)P• (cj)"'' (cj)P' · · · , (ck)P', (ck)"''• (cl)P; · · ·] i\JJk·
For convenience, we rewrite the above commutator as
=B!,J ,
(2.89a)
[(J)Jcj(jh , (k)Jcl(k).j i\!Jk ,
(2.89b)
where the symbol (j)t is a sequence of cJ and c1 labeled by j. Among the terms the above
commutator yields, we pick up the following term ,
That is why we called the operators BA and B,. (2.30) power-sum creation-annihilation operators . They also guarantee the mutual commutativity of the conserved operators,
(2.9-1)
f:(clrJ
k;J
~
~
~(ck)"J~ = s,.J~·
On the other hand, there is a commutator picked up form the Weyl ordered product (2.93) ,
(2.90)
of the Calogero model (2.12).
The final task in this section is to present a correspondence with the generalized Lax equations. By a straightforward calculation, we can confirm the following commutation relations:
[6i:,,cl]
2mw[(ck)P(ck)"'-tv'
(2.91a)
[6i:, , ck]
-2pw[(cl)P- 1 (ck)"'] w·
(2.91b)
[UJ1c}(j)2, (k)Jck(k)4]J\!Jk ,
which yields the following term,
(2.95)
From eq. (2.92), we notice that eq. (2.95) is the unique counter term of eq. (2.94). Thus we
confirm eq. (2.93). We can ,-erify the second formula (2.9lb) in the same way. The restriction
of eq. (2.91) generates the following equation that is similar to the generalized Lax equation
(2.52)
We shall verify the first formula (2.91a). Using eq. (2.68), we hm·e
N
-(1
L [[(cj)P(cj)"'Jw, cl]
:r: 1)pw ~ { [(L+JP- 1(£-l"'L} J~
j;J
+( 1 ± 1)mw
N
m
L [(c})P(cj)m-1 [cj, cl]Jw
j;1
N
]w- 2waJ.L- 1 L:[( (cj)P(c1)"'- 1 -
2mw[(ckJP(ck)"'- 1
(2.96)
The newly appeared operator-valued matrix, Ai:,, is defined as
(ckJP(ck)"'- 1
)MjkL·
N
);1
N
L(A!:,GhnJ =
n ::; J
From the identity,
[cJ,cl]
~{[(£+)P(£-)"'- 1 LL1
= -[c} , ck],
(2.92)
11.
2.:::(6~,- O~")Gk,.J
n=I
,
where G is an a rbitrary matrix that obeys
we can readily confirm the following identity,
(2.98)
(2.93)
\\"e can wrify that the .\~n matrices also satisfy the sum-to-zero condition. Substitution of the
unit matrix 1 into eq. (2.97) is possible because the unit matrix satisfies eq. (2.98):
which verifies the first formula (2.9la). Equation (2.93) is prm·ed as follows. Among the terms
in the Weyl ordered product, [(c})P(c1)"'M1kJw' we pay attention to the following term:
N
n=I
Since 6fnl =
On the other hand , there is a term in the other Weyl ordered product, -[(ck)P(ck)'"i\JjkJw'
which uniquely corresponds to the abo,·e term:
N
L (A!'n l )k,.J = 2.::: (6;,,-
(c} )P• (c1)'"' (c})P' · · · Jlf1k( c} JP'• (c1)"''• ( c} )P; - - .
(ck)P', (ck)"''• (ck)P; · · · JIJJk(ck)P• (ck)"'' (ck)P' · · ·.
(2.97)
11.
~
o~.J ,, '
11.
O~,) lk,.J
n=l
(2.99)
11.
the r.h.s. of the above equat ion vanishes. Then we get
N
2.:::(.1\;,,)k,.
n= l
= 0 k = l , ···,N.
(2.100)
38
CHAPTER 2. ALGEBRAIC STRUCTURE OF THE CALOGERO l'v!ODEL
Computation of the sum of the other suffix k needs a preparation. From eq. (2.98) , we ha1·e
t
1
J.L- I</mGknl
=
F/mGknflmJ.l-
t
Gknl·
1
I</ml~
k,n=l
JL
k,n.:::}
t
k,n ;:l
J1.
Thus the sum of all the elements of the matrix G is symmetric with respect to the exchange of
the indices. Then we get from the r.h .s. of eq. (2.97),
LN (6:;,- o:;,)Gkn I
k,n::::l
Jl
For an arbitrary operator-valued matrix G that meets the condition (2.98), the matrix Afn must
satisfy the following identity,
N
L
(A:;,)k/Gln = 0,
k,l,n=l
which means A!:, meets the sum-to-zero condition:
N
(2.101)
L(A:;, )kl = 0, I= 1, · · · , N .
k=1
Thus eq. (2.96) is rewritten as
E[O~n,Lt,Lnl~
=
,~[L±,A:'nL,l-(1'f1)pwE{[(L+)P- 1 (L-)'"JwL1
+(1 ± 1)mw E { [(£+)P(£-)m-1Jw}
k<L
(2.102)
and we conclude that the A~" matrices must be the i\1 -matri ces for the generalized Lax equations:
(2.103)
A:'n = 1\I:;,.
\Ve have clarified the correspondence between the quantum Lax and the Dunk! operator
formul at ions for the Calogero model. \\'e also ha1·e formulated a direct method to obtain the
generalized Lax equations from the commutator among the Dunk! operators. The correspondences we have considered a re based on the restriction
~~.
Thus any equality between a formul a
in the quantum Lax formulation and the correspond ing formula in the Dunk! operator formulation is a "weak" equality. A problem whether such equalities are "strong", or indepe nd ent of
the restriction, is also worth considering.
2.5. SUM.\JAR!'
2.5
39
Summary
We have im·estigated the algebraic structure of the quantum Calogero model in the framework of the quantum Lax formulation and the Dunk! operator formulation. From the Lax
equation for the classical model, we ha,·e obtained the Lax equation for the quantum Calogero
Hamiltonian (2.5) whose i\I-matrix satisfies the sum-to-zero condition (2.8). The sum-to-zero
condition has enabled us to construct a set of the conserved operators of the quantum Calogero
model, as ,,·as the case with the quantum Calogero-i\loser model [3-l, 74 , 75, 88, 89]. To show
the quantum integrability of the Calogero model, we ha1·e considered a construction of commuting consen·ed operators, eqs. (2.12) and (2.20). By use of the explicit forms of the first two
consen·ed operators, /1 and /2 (2.13), or 11 and 12 (2 .21 ), we ha1·e obtained the first two of the
generalized Lax equations for the commuting conserved operators (2.16) and have conjectured
the general form (2.22). However the generalized Lax equations for the commuting consen·ed
operators are not compatible with the recursi1·e construction. To study the recursi1·e construction of the generalized Lax equations, we ha1·e introduced a series of operators that are made
by sum of all the elements of the \\"eyl ordered product of the£+. and £--matrices (2.29).
From the explicit forms of the first few conserved operators (2.26) and power-sum creationannihilation operators (2.30), we have obtained the corresponding generalized Lax equations,
eqs. (2.28), (2.32) and (2.33). From these first few generali zed La:~: equations, we have found
out the recursion formulae, eqs. (2.36) and (2.53). Using the recursion formulae, we have recursi,·ely constructed all the generalized Lax equations (2.52). The generalized Lax equations
pro1·e the mutual commutativity of the power-sum creation-annih ilation operators, which will
play an important role in the algebraic construction of the energy eigenfunctions [76, 77]. \\"e
shall study this topic in Chapter 3. Defining the IV~'l-operators by eq. (2.60), we have prm·ed
that the generalized Lax equations yield the IV-algebra as a commutator algebra among the
w~·l-operators.
We have studied correspondences between the quantum Lax formulation and the Dunk!
operator formulation, which are de1·eloped independently in the study of the quantum Calogero
model [80]. The Dunk! operator formulation pro1·ides us a simple way of constructing the
commut ing consen·ed operators of the Calogero model (2.69). We ha1·e observed that the
restriction of the operand to the wa1·e functions of the id entical particles enables us to translate
the results in one of the theories into those in the other. To be concrete, we ha1·e related
arbitrary operators made from the two matrices, L + and L-, and their commutator algebras
with those in the Dunk! operator formulation, eqs. (2.73) and (2.76). A. method to directly
obtain the .\!.':.-matrices has been also obtained as eq. (2.97). :\lutual commutativity of the
consen·ed operators I, has been prm·ed.
In a sense, the quantum Lax formulation makes up for difficulties with the Dunk! operator
formulation and 1·ice 1·ersa. For example, "·e can readily see that the commutator algebra in the
quantum Lax formulation can be \\Titten in a closed form by the operators defined by (2.29).
Howe1·er, this is not observed in the Dunk! operator formulation. On the other hand, the Dunk!
operator formulation pro1·ides an easy way to prove the mutual commutat i1·ity of the consen·ed
operators of the Calogero model (2.69) and a direct method to construct the generalized Lax
equations, eqs. (2. 102). Howe1·er, in the quantum Lax formulation , a recursi1·e co nstruction of
the generalized Lax equations for the commuting consen·ed operators (2.22) remains unknown
because of the difficulty of the order problem of a pair of matrices, L + and L-. Thus the
translation between the two methods will help our deeper comprehension of the systems.
Chapter 3
Algebraic Construction of the
Eigenfunctions
Historically speaking, the Calogero model appeared as a one-dimensional quantum manybody problem with square and inverse-square long-range interactions [19]:
Hcalogero
Pi
=
=
(3.1a)
.8
-1-
(3.1b)
0Xj.
By a masterful separation of variables, Calogero solved the eigenvalue problem of the above
Hamiltonian in 1970. He found out a change of variables that yields an ordinary differential
equation for the eigem·alue problem of the Hamiltonian (3.1) , which is reducible to Laguerre's
ordinary differential equation. The formula of the energy spectrum has a similar form to that
for the N harmonic oscillators. The fact motivated Perelomov to try an a lgebraic treatment of
the system (3.1), though his challenge was not completed [61]. One of the aims of this chapter
is to complete Perelomov's approach using the quantum La' formulation.
Since the Hamiltonian (3.1) is translationally invariant, we may fix the center of mass at
the origin of the coordinate. This corresponds to the convention that we are not interested in
the motion of the center of mass, as is often the case with the statistical mechanics problem.
In this case, the Hamiltonian reduces to the following form,
He
(3.2a)
(3.2b)
which contains an external harmonic well instead of mutual harmonic interactions. This modification \ras introduced by Sutherland [70]. He obtained the exact ground state wave function
for the Ham iltonian and found out a correspondence with the theory of random matrices. Using
Calogero's energy spectrum formula, he also studied the thermodynamics of the above Hamiltonian (3.2). \\"e treat the Hamiltonian (3.2) instead of the original Hamiltonian (3.1). i\ote
that the center of mass is not always fixed at the coordinate origin. Only when we compare
41
42
CHAPTER 3. A.LGEBR..l.IC CONSTRUCTION OF THE EIGENFUNCTIO.VS
our results for the Calogero Hamiltonian (3.2) with those for the original Calogero Hamiltonian
(3.1), we fix the center of mass at the origin of the coordinate.
The other aim of this chapter is to consider the simultaneous eigenfunctions of the conserved
operators of the Calogero model, which must form the orthogonal basis. As a similar model
to the Calogero model, we know the Sutherland model [72, 73]. The Sutherland model is a
quantum many-body system on a circle with inverse sine-square interactions (1.21). Among the
recent works on the Sutherland model, the computation of the dynamical correlation function
of the model is worth mentioning [31, 32, 48]. A key role is played by the Jack symmetric
polynomials [49, 69], which form the orthogonal basis of the Sutherland model. The Jack
symmetric polynomials are also related to the singular vectors of the IV-algebra [9, 10 , 51],
which is the symmetry of the Sutherland model [3-!]. As we have seen in Chapter 2, the
Calogero model also has the !\"-symmetry structure [77]. However, in contrast to the Sutherland
model, we have no knowledge on the orthogonal basis of the model. It is expected to be an
essential tool for deeper comprehension of the Calogero model, such as rigorous calculations of
correlation functions and relationships with the representation theory of the IV-algebra. Those
expectations concerning the orthogonal basis of the quantum Calogero model motivate us to
challenge its construction .
As we have mentioned, we shall complete the algebraic construction of the energy eigenfunctions a Ia Perelomo,- and study the simultaneous eigenfunctions of the conserved operators of
the Calogero model. Factorizing the Hamiltonian into a pair of Hermitian conjugate operators,
we get the ground state energy and the ground state wa,·e function. By successive operations
of the power-sum creation operators obtained in Chapter 2 on the ground state wa,·e function ,
we construct all the energy eigenfunctions of the Calogero model (3.2). Fixing the center of
mass at the coordinate origin, we formulate the algebraic construction of the eigenfunctions of
the original Calogero Hamiltonian (3.1). On the algebraically constructed basis of the Hilbert
space of the Calogero model, we get a matrix representation of the second commuting consen·ed operator j 2 (2 .2lb). By a straightforward diagonalization of the matrix, we conjecture
an eigenvalue formula for the second commuting conserved operator and present the first seven
of the simultaneous eigenfunctions.
3.1
We demonstrate an algebraic treatment of the eigenfunctions of the quantum Calogero
model (3.2a). First, we shall identify the ground state wave function. We decompose the
Hamiltonian (3.2a) into the sum of the non-negative operators. From eq. (2.2la), we see that
the Hamiltonian is expressed as follows:
(3.3)
\Ve define a pair of Hermitian conjugate operators as
N
N
k;l
k;l
1
L Lti =Pi+ iwxi + ia L --,
Xk -Xi
k¢i
N
_
L Lik =Pi -
.
1WXi
+
l
.
N
Ia
k;l
L ---.
Xi- Xk
1
(3.5b)
k;l
k¢i
Using the abm·e operators, a new expression for the Hamiltonian (3.2a) is obtained:
(3.6)
N
Note that the operator
L h) hi
is a nonnegative Hermitian operator. Consequently, all the
i;l
eigenvalue of the Hamiltonian (3.6) must be larger than or equal to the ground state energy
1
-'
Eg = 2Nw(Na + (1- a)). It is interesting to observe that the ground-state energy Eg reduces
to that of N bosonic (fermionic) harmonic oscillators when we take the harmonic oscillator
limit a -) 0 (a -) 1). In these limits, the coupling constant g vanishes, as is easily recognized
from eq. (2.7). In order to obtain the ground state wave function, we ha,·e only to solve the
following equations:
hiiO) = 0, for j = 1, 2,-- · , N.
(3.7)
Namely we seek a state that is annihilated by the operator hi. The solution of eq. (3.7) is
expressed as the real wa,·e function of Laughlin's type :
IT
(xiO) =
(xj - xk)" exp(-
l<;j <k<;N
N 1
-wxj).
L
j;l
2
Bj =
(3.8)
B!, n =
TE(£+)" ,
[He, BJ] = nwBJ,
on the ground state. The abo,·e relations are given in eqs. (2.30a) and (2.3la). Let )..' denote
a set of positive integers less than or equal to N,
)..'
{.V"",(N-1)""- 1 , · ··,1" 1 }
~
'-_,----'
~
n,v
n.v- 1
UJ
(3.9)
T1:(~L+ L- + ~[L+ ,r-J )
+ 2Nw(Na + (1- a)).
(3.5a)
{:Y,· ·· ,N,N-l,··· ,N- 1,· ··,1 , ·· · ,1}
Changing the order of£+ and£- in the second term of the r.h.s. of eq. (3.3), we get
1
2TE(£+ £-)
43
Excited states are obtained by applications of the power-sum creation operators
1,2, · ·- ,N,
Perelomov 's Program
He
3.1. PERELOJ\IOV'S PROGRAM
(3.4)
\\"e call)..' dual Young tabl eau, which is the dual of the Young tableau. Later we shall introduce
a definition of the Young tableau. For details about the Young tableau and the dual Young
44
CHAPTER 3. ALGEBRAIC CONSTRUCTION OF THE EIGENFUNCTIO.VS
tableau, consult, for instance, ref. [50]. Here a function of the dual Young tableau l(X) is the
length of the dual Young tableau ,
3.2. SIMULT4.NEOUS EIGENFUNCTIONS
This formula means that the indices of the eigenfunctions of the original Calogero Hamiltonian
(3.1) should be the dual Young tableaux X that meet the following condition,
(3.10)
(3.15)
k=l
which is the number of non-zero elements in the dual Young tableau. The excited state labeled
by a dual Young tableau X and its energy eigenvalue E(X) are expressed as follows:
or in other words , n1 must be zero, n 1 = 0. Then the eigenfunctions are expressed as
N
l>.')calogero =
N
(3.1la)
k=l
(Eg
He IX)
+ IXIw)IX) ~ E(X)IX) ,
(3 .1lb)
1>-'1 =
:L knk = :L >-~.
3.2
k=l
[BA, Bl] = 0, n, m = 1, 2, .. -, as has been confirmed in Section 2.1. In the harmonic oscillator
BA is expressed as
BA a~O TE(diag(pl + iwxl,Pz + iwxz , ... • PN + iwxN)r
t
(3.13)
2JaJ),
n- 1,2,·
limit, a -t 0,
n
BJ
The correct formula for these operators was first gi,·en in eq. (2.30a) [76, 77].
(3.12)
Multiplicity of the n-th level for the system whose energy eigenvalue is Eg + nw is equa l to
the number of partitions p(n; N), i.e., the number of the different dual Young tableau X whose
weight is equal ton, #{XjiXI = n }. This is exactly the same result as is well known for the
N harmonic oscillators.
Independence of the states, which correspond to two different dual Young tableaux, X and
11-' respectively, can be confirmed as follows. First, we remark that the order of the creation
operators dose not matter because of the mutual commutativity of the creation operators,
N
B!.BJ
BA, n 2 5.
1(-1')
k=l
whose energy eigenvalue is E(X) = Eg + IXIw. This agrees with the result of Calogero [19].
Perelomov investigated an algebraic approach to the Calogero's system (3.1a) and succeed ed
in obtaining three kinds of "creation (annihilation)" operators , namely
and
(and
their Hermitian conjugates) by direct calculation. However, he did not give higher operators
where the symbol lXI denotes the weight of the dual Young tableau,
N
(3.16)
k=Z
II (Bk)"' IO) ,
1>-')
II (Bt)"' IO) ,
_
Simultane ous Eigenfunctions
\Ve have obtained an algebraic formula for the energy eigenfunctions of the quantum
Calogero model [76, 77]. Since the series of the eigenfunctions contains the same number of
independent elements as that of the non-interacting quantum harmonic oscillators, they span
the Hilbert space of the quantum Calogero model. Howe,·er, they do not form an orthogonal
basis of the Hilbert space because of the large degeneracy of its Hamiltonian. A com·entional approach to the construction of the orthogonal basis is the Gram-Schmidt method. The
Gram-Schmidt method compels us to compute many complicated multiple integrals and seems
hopeless. Instead, we shall consider the diagonalization of nontrivial higher conserved operators
of the model and remO\·e the degeneracy, as has been done for the wave functions of the hydrogen atom. The eigenvalues corresponding to the conserved operators are expected to uniquely
identify the elements of the orthogonal bas is obtained as simultaneous eigenfunctions.
From the study in Chapter 2, we know the commuting consen·ed operators of the Calogero
mode l [76, 80]. They take the forms
j=l
>.' and 1/ generate two different polynomials of the com·entional
(3.17)
creation operators for the quantum harmonic oscillators, {a}lj = 1, 2, · ·- , N}. Homogeneous
polynomials with degree n have independent elements whose number agrees with the number
of partitions, p(n; N). Thus we can get all the energy eigenfunctions algebraically by eq. (3.11)
and hence they form a basis of the Hilbert space of the Calogero model.
\\"hen we consider the original Calogero Hamiltonian (3.1), we must take it into account
where theN x N operator-valued matrices L +and L- are defined by eq. (2.6) in Chapter 2. The
Hamiltonian corresponds to the first conserved operator 11, i.e., 2Hc = 11. Thus we ha,·e a set
of independent operators that are simultaneously diagonalizable. The basis that diagonalizes
them all simultaneously is expected to be the orthogonal basis of the quantum Calogero model.
As a first step of the research of the orthogonal basis of the model , we shall diagonalize the
first nontrivial conserved operator 12 , and obsen·e how the degeneracy disappears.
First, we consider how to obtain the matrix representation of the operator 12 on the basis
given by the algebraic formula in the pre,·ious section [76, 77]. \\"e denote the ground state
of the model by IO) 1. The eigenfunction of the quantum Calogero model is labeled by a dual
Young tableau X (3.9). The corresponding eigenfunction is expressed as
Hence two different indices
that the center of mass is fixed at the origin of the coordinate. In this case, the operator
(and also B 1) reduces to zero:
B~
N
TEL+=:Lh!
k=l
t (-i:- +
k=l
UXk
N
iwxk)
= 0.
(3.14)
1>-')t
=
II (Bk)"'IO)t·
k=l
(3.18)
-
-
--------
CHAPTER 3. ALGEBRAIC CONSTRUCTION OF THE EIGENFUNCTIONS
46
The suffix 1 of the ket in the abo,·e expression means that the state fX)t diagonalize the first
conserved operator i 1 , or equivalently, the Hamiltonian of the Calogero model. The operator,
B~ = TE(L+)", is the power-sum creation operator. We recall that the eigenva~u e of the first
conserved operator, or equivalently twice the energy etgenvalue, of the state f..\ )t ts gtven by
[19, 76, 77]
(3.19)
where the symbol fXI is called the weight of the dual Young tableau (3. 12). Any ~wo states
with different weights are orthogonal. This means that a higher consen·ed operator I2 is blockdiagonalized by the energy eigenfunction basis and each block consists of the states of the same
weight.
In the calculation of the matrix elements on the basis (3.18), we shall utilize the fact that
the operator 0 A = TEAL-, where A is an arbitrary N x N operator-valued matrix, annihilates
the ground state, i.e., 0Af0) 1 = 0. \Ye use the generalized Lax equations obtained in Chapter
2 [76, 77] to derive the following formulae:
w2 N((a -1)- NarfO)t
i2]0)t
N
~ E2(0)f0)1,
n.-1
nw L (L +r+t L- + nw2a L BLB~-k
j,k=l
k=l
+nw2(2Na- (n + 1)(a- 1))B~,
(3.21)
2nmw 2 B~+m·
(3 .22)
] _
-
[
2
2w a
] [ ]2)21
E2(0)-4w 2(a-1)+4w 2 Na
f1 )t
E2 (0)- 6w 2(a -1) + 4w 2 Na
2
]
·
(3.23)
We have implicitly introduced above a lexicographic order among dual Young tableaux. That
is , two different tableaux X and p.' with the same weight are denoted by X > p.' if the first
non-vanishing dif!'erence ,\~ - p.~ is positi,·e.
Similar calculat ions lead us to t he (n+1)-th block of the operator i 2, M 2 (n), on the weight-n
7
basis in the lexicographic order denoted by D 1(n) = [ln) 1, fn- 1, 1) 1, · · ·, ]1")t] :
(3.24)
Diagonalization of the matrix J\!2 (n) gives the eigenvalue E2 (X) and t he corresponding eigenvector E 2(X). Then the following relation must hold among E2 (X), E 2 (X), J\!2 (n) and i 2:
i2T,~ D 1(n)
1
T,.
-
1
E2(..\) = 4w
2(
1
(
4N (a -1)- Na
)
2
I(A')
+ (1- a+ 2Na)fXf- a {;(..\~f +
N
)
{;(..\d
.
(3.28)
Here, Ak is the k-th element of the Young tableau..\, which is the dual of the dual Young tableau
X defined by
..\k
N
Lni,
j=k
(3.29a)
nk = ,\k- ..\k+t, AN+t~O.
(3.29b)
Namely, the dual Young tableau X is obtain ed by exchanging the rows and the columns of
the Young tableau ..\ and vice versa. The eigenvalue E2 (X) removes most of the degeneracy.
Indeed, there ts no degeneracy up to the states with weight 5. We can obtain the expresstons for the orthogonal eigenfu~1ctions fron:_ the simu ltaneous eigenvectors corresponding to
the nondegenerate etgenvalues £ 1 (X) and E2 (X). However, there still remains deaenerac,·.
For example, two pairs o~ dif!'ere!1t tableaux, {4, 12}, {3 2 } and {3, 13}, {23} respecti:ely gi,~e
t_!1e s~me etgem:alues of It and h. Since ~qs. (3.19) and (3.28) suggest that the eigem·alues
E,.(..\) of the htgher conserved operators I ,., m ::=: 3, involve the higher-order power sums of
I
the Young tableau, p,.(..\) = 2:(..\k)"', remaining degeneracy must be completely removed b,·
k=l
-
their diagonalizations.
Generally speaki ng, the eigenfunctions of the quantum Calogero model (3.18) are gi,·en
by products of symmetric polynomials and the ground state wa,·e fun ction. We denote the
symmetric-polynomial parts of the energy eigenfunctions by
[X],~(xf..\'),
(xfO), ·
The meaning of the suffix 1 in the above expression is the same as that of the suffix for the
ket fX)t. The coordinate representation of eigenfunction (3 .18) is expressed in terms of power
N
sums p,. (x,, · · · ,xN)
=L
xz. The first se,·en eigenfunctions are
k=l
1
T;;' M2(n)TnT,.- D 1 (n)
diag[E2(n), E2 (n -1, 1) , · · ·, E2 (1")]T,~ 1 D 1 (n),
[E 2(n), E 2 (n, n- 1), · · ·, E 2(1")].
47
where the suffices 2 of the kets in the above ex_rression mean that the state f..\') 2 diagonalizes
the first two commutmg cons_erv~d operators, I 1 and i 2 . In particular, the elements of D 2 (n)
that have no degeneracy m £2(..\) turn out to be the elements of the orthogonal basis of the
model.
After a straightforward but lengthy calculation, we can obtain the eigenvalues of j 2 for up
to the seventh block (corresponding to weight 6). The eigenvalue E2 (X) is expressed as
(3.20)
By using the above formulae, we can express the operator i 2 in the form of a c-number-valued
matrix. For instance, the block with weight 2 is expressed as
j [ ]2) 1
2 ]1 2)
1
3.2. SIJ\IULT4.NEOUS EIGENFUNCTIONS
(3.25)
[0],
= 1,
(3.30a)
(3.26)
[1] 1
= 2iwp 1 ,
= -4w 2 p2 + 2wN + 2wN(N- 1)a,
(3.30b)
\\'e denote by ]X)2 the weight-n states that diagonalize the operator i 2. The equality (3.25)
shows us their forms as
(3.27)
[2],
(3.30c)
[1 2] 1 = -4w 2(pt) 2 + 2wN,
[3] 1 = -8iw 3p3 + 12iw2 ( 1 + a(N
(3.30d)
-1)) p
1,
(3.30e)
48
CHAPTER 3. ALGEBR<\IC CONSTRUCTION OF THE EIGENFUNCTIONS
[2 , 1)t
=
-8iw3 p2p 1 + 4d((!V
[1 3 )t = -8iw 3 (p 1 ) 3
(3.30f)
+ 2) + aN(N -l))pt ,
+ 12iw2NPt·
3.3. SUMMARY
49
first seven orthogonal symmetric polynomials (3.31) are cast into the foll owing forms,
[0] 00 = 1,
(3.30g)
[1] 00
Note that coefficients of the power sums depend on the particle number N and the parameter
constant. We denote polynomial parts of the s1multaneous
a th a t determines the coupling
_
eigenfunctions for 11 and !2 in a similar way by
=
(3.3-la)
2iwm1,
[2] 00 = 8w
2(
m1'
(3.3-lb)
aN(N-1))
+,
2w
2
2
2
[1 ]00 = -4w ((1
[Xh = (xiA2)2.
[3]00 -_
(x!O)!
-
48 .
1w
3(
+ a)m2 + 2am 1' - .!_N(Na + 1))
mp
3
Then the first seven of them are expressed by the basis [X)t as
[2, 1]oo = 8iw ((2a
(3.31a)
= [D)t = 1,
= [1] 1 = 2iwpt ,
2
[2h = [2)t- [1 2)t = -4w 2 (p 2 - (pt) ) + 2waN(N- 1) ,
[12h = [2)t + a[1 2] 1 = -4w 2 (p 2 + a(p 1?) + 2wN(Na + 1) ,
[0]2
[1
(3.31b)
[1]2
3
] 00
=
3
-8iw ((a
2
+
2w
1 (N- 1)(N- 2) )
wa
m1 ,
2
2
+ 3a+ 2)m3 +3a(a+ 1)m2,
1
+ [1 3 ]1
2
3p2p 1 + (pt) 3 ) - 12iw a(N- 1)(N- 2)p1 ,
(3.3-ld)
'
(3.3-le)
.
+ 1)m2,1 + 6am 1, - ;j-(1a)(N_w
1)(Na + 1)m 1) ,
+6am 1, -
2~(a N
2 2
(3.3-lf)
+ 3aN + 2)m
1
) ,
(3.3-lg)
(3.31c)
(3.31d)
(3.3-lc)
where m>. denotes the symmetrized monomial, or in other words, the monomial symmetric
polynomial whose definition is
[3h = 2[3)t- 3[2, 1)t
= -8iw3 (2p 3 -
(3.31e)
CTESN, distinct
permutations
3
[2 , 1]2 = [3]1- (1- a)[2, 1)t- a[1 )t
= -8iw 3 (p 3
-
(1- a)p 2p 1 - a(pt) 3 ) - 4iw2(1- a)(N- 1)(Na
+ 1)p1 ,
(3.3lf)
[1 3 ]2 = 2[3)t
= -8iw
3
+ 3a[2, 1)t + a 2[1 3 )t
(2p 3
+ 3ap2p1 + a2(p 1 ?) + 12iw 2(a 2N 2 + 3aN + 2)P1·
(3.31g)
Since the eigenvalues E1 (A') and E2 (X) for the above seven symmetric polynomials have no
degeneracy, they are indeed the simultaneous eigenfunctions of all the commut tng, conserved
operators, 1n and hence In, n = 1, 2, · · ·. In this sense, they should be denoted by [A loo · These
are the first seven of a new series of ort hogonal symmetnc polynOimals w1th respect to the
inner product:
N
{" II dxk II
lxk-
xd 2" exp( -wx 2)[A']oo[l/]oo
1'3!k<l-:!N
-oo k=l
(3.32 )
Nx6>.·~··
Thus they should be regarded as a deformed multivariable generalization of the Hermite polynomials.
\\"e notice that the eigem·alue formula for the commuting conserved operator h (2.13b),
N
E2 (A)
= 4w 2 L ((Ad+ a(N + 1 -
2k)Ak),
(3.33)
k=1
"·hich can be deriwd from eqs. (2.13) , (2.21), (3.19) and (3.28), has the same form as that of
the eigenvalue of the Sutherland Hamiltonian on the Jack polynomial [69 , 73]. llloreover, the
The above expansions show the triangularity, which is also observed for the Jack polynomials.
The meaning of this observation will be discussed in more detail in the next chapter. These
observation strongly suggests that there must be a similarity between the Jack polynomials and
the unidentified orthogonal symmetric polynomials associated with the Calogero model. To be
concrete, the observation of the eigem·a!ue formula (3.33) and the triangularity (3.34) sho"· the
essential part of the definition of the simultaneous eigenfunctions of all the commuting consen·ed
operators of the Calogero model. Based on the results of this section, "·e shall introduce the
definition and investigate the properties of the simultaneous eigenfunctions in the next chapter.
3.3
Summary
We ha,·e studied an algebraic construction of all the eigenfunctions of the Calogero Hamiltonian with the help of the quantum La' formulation. By the factorization of the Hamiltonian ,
we have obtained the ground state "·a,·e function.
sing the power-sum creation operators
which has been obtained in Chapter 2, we ha,·e formulated an algebraic method to construct
the eigenfunctions of the Calogero model. From the number of independent eigenfunctions,
we have confirmed that the eigenfunctions form the basis of the Hilbert space of the Calogero
model. \Ve ha,·e also reproduced the result for the original Calogero model by fixing the center
of mass at the coordinate origin. Thus we have completed Perelomov's dream of the algebraic
construction of the eigenfunctions of the Calogero model. \\'e have also considered a construction of the orthogonal basis of the Calogero model by diagonalizing mutually commutin_g
conserved operators. \Ve have direct ly diagonalized the first nontrivial consen·ed operator !2
using the energy eigenfunctions with weights up to 6. The results indicate a general formula
for the eigenvalue of 12. In addition , we ha,·e presented explicit expressions of the first se,·en of
------------
--
50
CHAPTER 3. ALGEBRAIC CONSTRUCTION OF THE EIGENFUNCTIONS
the unidentified orthogonal symmetric polynomials associated with the Calogero model. Fr~m
· farm o f tl1e 11-e 1·ght function of the inner-product , we ha1·e concluded
the exp I.JCJt
.
. that the se1. en
1
·
·fi ed art hogona1 symm etric polynomials should be regarded as multJvanable
umdentJ
. genera Jza·
f h H
"te polynomials From the eigenvalue formula for the commutmg conserved
tJons o t e erm1
·
.
· 1
·
xpansion of the explicit forms w1th respect to the monom1a symmetnc
operator I 2 an d the e·
· 1
d 1
"d
· fi d
·
1
b e ved a similarity between the Jack polynom1a s an t 1e um ent1 e
functwn, we 1ave o s r
.
.
d 1 "d
·fi
· olynom"1ais This observation give us a cruc1al hmt towar t 1e 1 ent1 orthogona I symme t nc P
·
.
.
0
·
.
f the orthogonal sy·mmetric polynomials assoc1ated w1th the Calogero model. etalied
ca t Jon o
. d fi . ·
·
t fa m
d
mathematical properties of the polynomials , such as the1r e 111t10n m a compac
r an
their formulation , are to be investigated in the next chapter.
Chapter 4
Orthogonal Basis
Exact solutions for the Schriidinger equations have provided important problems in physics
and mat hematical physics. i\lost of us have studied the Laguerre polynomials and the spherical
harmonics in the theory of the hydrogen atom, and the Hermite polynomials and their Rodrigues
formu la in t he theory of the quantum harmon ic oscillator. The former is also a good example
that s hows the role of conserved operators in quantum mechanics. The hydrogen atom has
three, independent and mutually commuting conserved operators, namely, the Hamiltonian ,
the total angular momentum and its z-axis component. The simu ltaneous eigenfunctions for
the three consen·ed operators give the orthogonal basis of the hydrogen atom. A classical
system with a set of independent and mutually Poisson commuting (im·olutive) consen·ed
quantities whose number of elements is the same as the degrees of freedom of the system can be
integrated by quadrature. This is guaranteed by the Liouville theorem. Such a system is called
the comp letely integrable system. Quantum systems with enough number of such consen·ed
operators are analogously called quantum integrable systems. The hydrogen atom is a simple
example of the quantum integrable system.
Among the var ious quantum integrable systems, one-d imensional quantum many- body systems wit h im·erse-square long-range interactions are nOll" attracting much interests of t heoretical
physic ists. Of the 1·arious integrable inverse-square-i nteract ion models, the quantum Calogero
model [19] has t he longest history. Its Hamiltonian is expressed as
(4.1)
where the constants N, a and w are the particle number, the coupling parameter and the
strength of the external harmonic well respecti1·e!y. The momentum operator Pi is giYen by
a
a partia l differential operator , Pi = - i - . I n Chapter 2, 11·e have confirmed that the model
OXj
is a quantum intearable s1·stem in the sense that it has sufficient number of independent and
mutually commut~1g con;e JTed operators [62, 76- 78 , 80]. On the other hand , the Sutherland
mode l [72, 73], wh ich is a one-dimensio na l quantum integrable system with inverse-sine-sq uare
interactions,
1 N 2 1 N
a2 - a
Hs = - ""'P1· + 2
,
2 J;J
L
2 J.,k;J sin (x 1· - X k)
L
j#k
51
(4.2)
'
~--------------------
~
--
CHAPTER 4. ORTHOGON.-I.L BASIS
52
has been t horoughly invest igated since its orthogonal basis was kn ow n to cons ist of the J ack
symmet ric polynomials [36, 50, 69]. T he quant um Lax formul atw n a nd t he Dunk! operato r formulat ion (exchange operator fo rmalism) showed t hat t hese t wo models sha re the same algebrmc
structure [13 , 14, 34, 35, 62, 76-78, 80]. We have a lso observed in C hapter 3 that the first se ven
simultan eous eigenfunct ions of the commuting conserved opera tors of t~1e Calogero mod el and
t he Jack polynomi als have a co mmon property, na mely the t n a ngulan ty w1th respect to t he
domin ance order [69] . T hese facts st rongly suggest so me s imil arities in the stru ctures of the
Hilbert spaces of t he Calogero a nd t he Sut herla nd models. In order to cl anfy th1s problem , we
shall apply a naive a pproach t hat we use in t he st udy of the hydrogen atom t o th e quantu m
Calogero model and study t he deformed mult ivaria ble extension of t he Herm1te polyn01ma ls,
namely, t he Hi-J ack (hidden-J ack) symmet ric polynomia ls [79, 83, 84, 86, 87]. .
.
The J ack symmetric polynomials are uniquely determined by t hree propert1es [50, 69]. F mt ,
they are t he eigenfunctions of th e differential opera t or th a t is derived from the Hamiltonian
of the Su t herla nd model. Second , th ey possess triangula r expanswns 111 monom1al sy mmetnc
polynomials with respect to the dominance order. And last, t hey a re properly norm alized. For
detail , see eq. (4.38). Quite recently, Lapointe and Vinet discovered the Rodrigues formul a
for the Jack sy mm etric polynomi als using the Dunk! op erat or for th e Suth erland model [4 2] .
Here we shall extend t heir results to t he qu antum Calogero mod el and give the Rodrigues
formula for the Hi-J ack sy mmetric polynomials [83, 84] . Through th e Rodrigues formu la , we
shall also investigate the properties of the Hi-Jack sy mmetric polynomia ls and compa re th em
with the basis of the model that was given by th e qu ant um Lax formula tion [76 , 77] in Ch apter
3 and by the Dunk! operators [17, 18] . The algebraic const ruction of t he eigenfun ctions for t he
Hamiltonian (t he first conserved operator) of t he qu ant um Calogero model h as already b een
given. Thus the Hi-J ack symmetric polynomials must b e linear combina tions of them . We shall
specify the linear combin ations t hat relate the Hi-Jack polynomia ls and the eigenfun ctions of the
Hamiltonian given before [17, 18, 76, 77] . We also want to see the relationships and simila ri t ies
between the J ack polynomia ls and the Hi-Jack polynomials. A proof of the ort hogona lity of
t he Hi-Jack polynomials is also an important problem. We shall present clear answers t o these
questions.
The outline of this chapter is as follows. In Section 4.1 , we summarize and reformul ate
the res ults of th e qu a nt um Lax and the Dunk! operator formulation s for th e C a loge ro and
Sutherland models. We shall clarify the common algebra ic stru cture of the models. The Jack
symmetric polynomials a re a lso int roduced . In Sec tion 4.2, we prese nt t he Rodrigues formu la for
t he Hi-J ack symmetric polynomials and int roduce som e propos itions that guarantee the resul ts.
Some propert ies of the Hi-J ack polynomials such as integrali ty, t riangularity, orthogo na lity
and relat ionships wit h t he J ack poly nomia ls are also prese nted. In Sect ion 4.3 , we prove t he
proposit ions. And in t he final section, we give a brief summa ry.
4 .1
Mode ls and Formulations
In our deri1·at ion of the Rodrigues formula for th e Hi-Jack sy mmet ric polynom ia ls, we do
many computat ions im·olvin g the Dunk! op erators for th e Caloge ro model. We a lso compare
t he Hi-J ack polynomi a ls wi t h t he Jack polynomi als and wi t h the bas is of the Cal oge ro model
given by t he quant um La:x form ulation in Ch apter 3. Thus we need a summary of the Calogero
model, t he Sut herland model, and t he qu a ntum Lax a nd t he Dunk! operat or formulati ons.
-i .l . MODELS .-\ .'ID FORMUL.-I.TIO.\iS
53
First, we reform ulate t he q uantum Lax formulation and the Dunk ! operators for the Calogero
model. The Lax mat n ces for theN-body system are gi1·en by N x N operator-valued matrices.
To ex press t hem , we ha1·e to introduce two operat or-valued matrices:
(-L3a)
(-1 .3b)
where j, k ~ 1, 2, · · ·_ , N . T he above L-mat rix is for the (rational) Calogero-:d ose r model (1.20)
whose Ha m!l to ma n lS obtamed by taking w = 0 of the Calogero model (4. 1) or t he rational
limi t of th e Su t herland model (4.2). The Lax matrices for t he Calogero model are
L-
L +wQ ,
( -l A a)
'+
1 --- (L - wQ).
(4.-lb)
L
=
9_w
In eqs. ( 4.3) a nd ( 4.4) above, we have in t roduced different normalizat ions from t hose mat rices
in Chapter 2. We have a lso in t roduced accent ma rks. They a re just for con venience of late r
disc uss ions in t his cha pter . T hen t he Ha miltonian (4.1 ) is ex pressed by the above Lax matrices
as
wTEL+L-+ ~ Nw(Na+ (l -a) )
He
wT EL+ L-+Eg,
(-1.5)
N
where T E denotes a s um of a ll the matrix elements , T E A =
L
Aij , a nd Eg is t he ground state
i,j=l
energy. Note t hat t he first term of the r. h. s. of eq. (4.5) is a no nnegati1·e Hermit ian operator.
T hus t he ground state is t he so lution of t he following eq uat ions,
N
L
Ljk¢g =
0, for j = 1, 2, · · · , N => ifc¢g = EgJ,g.
(-16)
k=l
The ground state wa,·e fun ct ion is the real Laughlin wa1·e fun ction:
¢g=
IT
1
N
-
j= l
lxj - xklaexp(-?w 2: xJ).
l$j<k$N
(-1. 7)
A short note might be in order. Th e phase of the di ffe rence produ ct of t he above rea l La ughlin
wa1·e fun cti on, which determ ines t he statist ics of t he pa rt icles, or in other words, t he svmmet n ·
of all t he eigenfun ctions, can be a rbit rary. \Ye can assign a ny phase factor to all t he e~ch ang;s
of pa rt icles. Howe1·er , we must in t rod uce a phase facto r to t he d efi nit ion of t he Dunk! operators
[80], as has been disc ussed in Cha pter 2. To am id unnecessa ry com plex ity, 11·e fix t he ph ase at
un ity.
Th e eige nfunc tion of th e Caloge ro model is fa torized into an inhomoge neous sy mmet ri c
polynomial a nd th e ground sta te wa1·e fun ction. Fo r co nve ni ence of im·estigations on the
inh omoge neous symmetric polyn omia ls, we redefin e t he Lax mat rices (4.4) by th e follmrin g
simil ari ty tra nsform a tion:
o~ 1 L-¢g·
(-!.Sa)
og·
(-l .Sb)
d>~ 1 L+
CHAPTER 4. ORTHOGONAL BASIS
54
Any operator with a hat,
6,
is r~lated to an operator 0 by the similarity transformation using
4.1.
MODELS AND FORMULATI01VS
where AN+I
55
=
0 and the eigenva lue for the first consen·ed operator is
the ground state wave function </>g,
0
(4.9a)
6
(4.9b)
A set of mutually commuting conserved operators of the Calogero model
is given by
Unln
= 1, 2, · · · , N}
(4.10)
The Hamiltonian He is equal to w! 1 + Eg· We regard the first conserved operator J, as the
Hamiltonian of the Calogero model. The Heisenberg equations for the L- and L + matrices are
expressed in the forms of the Lax equation [76]. 1\loreo,·er, we have more general relations for
a class of operators,
(4.17)
It is nontrivial that.</>,~ is indeed an inhomogeneous symmetric polynomial. We shall prO\·e
It Ill Section 4.~. N?te that the eigenfunction of the original Hamiltonian He (4.1) and its
eigemalueaie </>,~ = </>g</>;. and wE, (A)+ E 8 . These eigenfunctions gi,·e the complete set of the
eigenfunctiOns. However , they are not orthogonal because of the remaining large deuenerac,·.
Usmg the Dunk! operator formali sm, we can do an analogous im·estigation on th; Calog~ro
model. The Dunk! operators for the model are
(4.11)
(.USa)
where the subscript w means the vVeyl ordered product. The class of operators naturally
includes the Hamiltonian, He= wV11 . The generalized Lax equations are
(4.18b)
[vPm' L-]
[L-,
Z;']- p[(L-)m(L+)P-']w ,
(4.12a)
[vPm,L+]
[L+ ,z;] +m[(L-)m-I(L+)P]w,
(4.12b)
where the symbol Z;' is an N x N operator-valued matrix that satisfies the sum-to-zero
condition:
N
N
'L,(Z;')ik = 'L,(Z;')ki = 0,
j=I
fork= 1, 2, · · ·, N.
(4 .13)
(.USc)
where Ktk is the coordinate exchange operator. The operator J(1k has the properties
Ktk = Kkt, (Ktk? = 1, K1~ = Ktk. K 1k · 1 = 1,
KtkAt = Akf<tk, f<tkAi = AiJ(Ik, for j # l, k,
(4.19)
j=I
As we have confirmed in Chapter 2, the generalized Lax equations exhibit that the operators
(4 .11) satisfy the commutation relations of the lV-algebra [77, 78].
T he operators with m = 0 are important in the construction of the eigenfunctions of the
Ham iltonian , because they sat isfy
where Aj is either a partial differential operator
_!!__
EJx.
(or equivalentlv a momentum operator
J,
or a coordinate excha;ge operator J( .k k = 1 2 ... N k ...t 1·
1'
' ,
,
,
r .
e abO\·e properties of the coordinate exchange operator are also expressed as the action on
a multi,·ariable function:
Pi),
Th a particle coordinate
Xj
(4.14)
(I<tkf)(x,, ... ,Xt, ... ,Xk, .. · ,XN) = j(x 1, • • • ,Xk, · · · ,x1, • • • ,XN)·
Thus the operators VP0 play the same role as the creation operator in the theory of the quantum harmonic oscillator. We call these mutually commuting operators VP0 power-sum creation
operators, whose meaning has been explained in Chapters 2 and 3.
Successive operations of the power sum creation operators generate all the eigenfunctions
of the Hamiltonian , which are labeled by the Young tableaux. The Young tableau A is a
non-increasing sequence of N nonnegati,·e integers:
(4.15)
(\',~);."( I~L ,);.,v_,-;.,v
... (l',o)-1,--1, . 1
A"
II (VpO),\;-,\k+l
p=I
.
1,
Note that the action on the ground state of the above Dunk! operators has already been remO\·ed
by the similarity transformation (4.9) . The Dunk! operators satisfy the relations,
(al, Om)
= 0,
[aJ, a!nJ = 0,
(4.16)
(4.2la)
N
[at, a;,] = Otm( 1 +a L I<tk) - a(1- OtmWtm,
(4.2lb)
k=I
kfl
[dt, dm] = a(dm- dt)Ktm,
O't·1 = 0.
Then the polynomial part of the excited state </>;. is given by [76, 77]
(4.20)
(4.2lc)
(4.2ld)
:\.s we have mentioned, the phase factor of the difference product part of the ground state wa,·e
function can be arbitrary. This phase factor affects the definition of the Dunk! operators and
coordinate exchange operators with hat , i.e. , 1,
d1 and ktk· \\'e ha,·e to introduce a phase
a aJ,
56
factor in the defining relations of the coordinate exchange operators (4.19), the action on the
multivariable functions (4.20) and the commutation relations of the Dunk! operators (4.21),
as we have seen in Chapter 2 [80]. This modification is naturally introduced by the inverse of
the similarity transformation of the Dunk! operators (4.9b). Using the above relations, we can
confirm that the eigenfunctions of the Hamiltonian h are given by [17, 18]
(o:~<l)l>.' (o:~<2/' · · · (o:~(N/" · 1
I:
a:ES,v
4.1. lviODELS AND FORMULATIONS
57
where Eg is the ground state energy. As is similar to eq. (4.6), the ground state of the Sutherland
model satisfies the following equations ,
N
"'L
L.., 1 k?fig =
k=1
0, for J. = 1, 2, · ·. ' N => H-S'i'g.i. - •'g ,!J
-r g >
because the first term of the r.h.s. of eq. ( 4.28) is a nonnegative operator. The ground state is
given by the trigonometric Jastraw wave function:
dis~inct
permutat•on
m>.(o:l, at · ··, o:~v) · 1,
(4.22a)
E1(>.)cp>.,
(4.22b)
?fig= II
lsin(xj-Xk)l".
(4.30)
1$j<k$N
The phase factor of the above trigonometric Jastraw wa,·e function can be arbitrary. By the
change of the variables ,
where m>.(x 1, x 2, · · ·, XN) is a monomial symmetric polynomial [50 , 69] defined by
(4.23)
o-ES,v: distinct
exp 2ixi
= Zj,
j
= 1, 2, . . . , N,
(4.31)
the Hamiltonian of the Sutherland model (4.2) is transformed to
permutation
Note that the summation over SN is performed so that any monomial in t he summand appears
only once. In terms of the Dunk! operators, we can express the commuting conserved operators
In (4.10) and the operators
(4.11) as
(4.32)
vpm
N
~(dt)"lsym'
I,.
(4.29)
n = 1, 2, · · ·, N,
(4.24)
I
11·1ere
p, . =
J
.
a
-1-.
azj
T
he ground state wave function (4.30) is transformed to
N
v.m
=
~[(atlm(o:))PJwjsrm'
(4.25)
.i.
'l'g =
N
II Iz1. _ Zk I" II Zj
-ta(N-1)
1'5j<k$N
where the symbol
means that the operand is restricted to symmetric functions [80] . Then
J
Sym
the power sum creat ion operators are expressed by
.
(4.33)
j=l
Here we do not mind the difference of the scalar factor of the ground state wave function. The
similarity transformation of the above Hamiltonian yields
(4.26)
(4.3~)
where Pn (x 1 , x 2, · · · , XN) is the power sum symmetric polynomial of degree n [50, 69]. This
shows that two kinds of eigenfunctions (4 .16) and (4.22) are related by the transformation
between the power sum symmetric polynomials and the monomial symmetric polynomials.
Next, we consider the quantum Lax and the Dunk! operator formulations for the Sutherland
model. The Lax matrix for the Sutherland model is
The above projected Hamiltonian can be deri,·ed from the Lax matrices (4.3). \\"e define the
"ground state" for the L-matrix ( 4.3a) by the solution of the equations,
N
L
(4.27)
The abo,·e Lax matrix gives the Hamiltonian of the Sutherland model by
Ljkt:;•
= 0,
for j
= 1, 2, · · · , N,
(4.35a)
k=1
and their solut ion is
(4.35b)
1$j<k$N
(4.28)
The phase of the abO\·e Jastraw wa,·e function also can be arbitrary. The effect of the phase
to the Dunk! operators for the Sutherland model, which can be made explicit by the similarity
58
transformation, is also the same as that of the Calogero model. By the similarity transformation
using the above Jastraw function, we define L and Q as
L
= t;-•Lt;•,
(4.36a)
Q
= t;-•(Jt;•=Q.
(4 .36b)
4.1. MODELS AND FOR.\IULATIONS
59
Since eqs . (4.12) and (4.42) has the same form , we notice the correspondence between the
quantum Lax formulation of the Calogero model and that of the Sutherland model:
(4.45)
(4.37)
This means that the two quantum Lax formulations for the Calogero and Sutherland models
give two different representations of the same commutator algebra.
The same situation can also be observed in the Dunk! operator formulation. \\"e introduce
the Dunk! operators for the Sutherland model whose action on the ground state is remo,·ed in
a similar way to deal with the Dunk! operators for the Calogero model:
From now on, we take I 2 as the Hamiltonian of the Sutherland model. The Jack symmetric
polynomials J>.(x ; 1/a) are uniquely defined by [59]
(4A6a)
Then we get the projected Hamiltonian (4.34), whose variables are not {zi} but {xi} by
Hs-
2TE(QLj2
tg
N
IzJ>.(x; 1/a) = :L(Ak + a(N + 1- 2k)Ak)l>.(x; 1/a) ,
(4.38a)
(4.46b)
(4.38b)
(4.46c)
k=l
=L
J>.(x; 1/a)
V>."(a)m"(x),
These Dunk! operators satisfies the following relations,
0
1-'~).
vu(a)
= 1,
(4.38c)
['VI, 'V m] = 0,
[x1, Xm] = 0,
D
where x = (x 1,x2, ... ,xN) and A and J1. are the Young tableaux (4.15). The symbol
dominance order among the Young tableaux [50, 69]:
::0
is the
(4A7a)
N
['VI, Xm]
= 6/m( 1 +a L
J(lk) -
a(l- 6/m)K/m,
(4.47b)
k=l
k;tl
D
N
J1.9
¢'}
L
N
L
Jl.k =
k=l
l
Ak and
k=l
l
L
Jl.k:::;
k=l
L
(4.39)
Ak for alii .
k=l
Note that the dominance order is not a total order but a partial order. A total order among
the Young tableaux is given by the lexicographic order:
N
J1. :::; A ¢'}
L
k=l
[Dt, Dm] = a(Dm- Dt)Ktm ,
( 4.4 7c)
\?1 · 1 = 0,
(-L47d)
which are completely the same as those of Dunk! operators for the Calogero model (4.21).
Commuting conserved operators (4.41) are expressed by the Dunk! operator as
N
Jl.k
=L
>.k and the first non-vanishing difference >.1 - J1.1
> 0.
(4.40)
N
In = L(Dt)"l
k= l
l=I
, n = 1, 2, · · ·, N.
(4.-!8)
Sym
Commuting conserwd operators of the Sutherland model are given by
(4.41)
In= TE(QL)".
We have similar relations to the generalized Lax equations for the Calogero model (4 .12) ,
[L, Y~j- q[L 1Qq- 1]w ,
(4.42a)
[Q, Y~] + m[L1Qq]w,
(4.42b)
where the operator u~ is defined by
u; = TE[L Qq]w.
1
(4.43)
The operator-Yalued matrix l~ also satisfies the sum-to-zero condition,
N
p.:
:L(l '~)ik = :L(l'~)ki
j=l
i=l
= 0,
for k
= 1, 2, · · · , N.
(4.4~ )
Thus we notice the correspondence between the two sets of Dunk! operators:
(4A9)
i\loreover, in the limit w -7 oo, the Lax matrices and the Dunk! operators for the Calogero
model reduce to those for the Sutherland model. Thus our theory for the Hi-Jack symmetric
polynomials described by the Dunk! operators for the Calogero model contains the results for
the Jack symmetric polynomials written by the Dunk! operators for the Sutherland model.
\Ve have summarized the quantum Lax and the Dunk! operator formulations for the Calogero
and Sutherland models. They give two different representations of the same commutator algebra. The quantum La, formulation and the Dunk! operators for the Calogero model include
those for the Sutherland model as a special case w -7 oo . Thus we can say that the theory
of the Calogero model and the Hi-Jack polynomials is a one-parameter deformation of that of
the Sutherland model and the Jack polynomials. In the following section, we im·estigate the
Hi-Jack polynomials using the Dunk! operators.
CHAPTER 4. ORTHOGONA L BASIS
60
4 .2
4.2. HI-JA CK SYMJ\IETRJC POLYJ\.OMIALS
61
where J is a s ubset of a set {1, 2, · · ·, N} whose number of elements
{1 ,2, · · · ,N }, IJI = k. From eq. (4.21c), we can ,-erify a n identity,
Hi-Jack Symmetric Polynomials
Followin g t he defi nit ion of the J ack symmetric po lynomials (4.38), we defin e t he Hi-J ack
(d; + ma)(di + (m + 1)a)
symmetric polynomials j,( x; w, 1/a).
{iJ)
lSym
= (di
IJI
+ ma)(d; + (m + 1)a)i
is equal to k, J ~
{i j)
' ,
Sym
(4.53)
Definition 4. 1 The Hi-Jack polynomials j >, (x ; w, 1/a) a1·e uniquely specifi ed by th e f ollowing
where m is an intege r. T he symbol
four conditions:
.
N
I 1j,(x; w, 1/a)
I: >.k]>.( x ; w, l /a)
k=l
(4.50a)
E 1 (>.)j>.( x ; w, 1/a),
.
IJ
where J is some set of in tegers means t hat the operand
Sym
IS res tnctecl to th e space o f t he mu lti variable fu nct ions t hat a re symmetric with respect to the
excha nges of a ny mcl1ces Ill the set J. T his ident ity gua rantees t hat the operator d
does not
depe nd on t he order of t he elements of a set J. Th e ra ising operators of t he Hi- Jack ;~lynomials
are expressed as
N
I:(>.%+ a(N + 1 - 2k)>.k )J>.( x ;w, l /a)
I:
j,( x ; w, 1/a)
w,~(a , 1 /2w)m~(x),
Here ).
= {>. 1 ~ >. 2 ~
(4 .50cl)
1.
··· AN ~ 0} E YN is a Young tableau and YN m eans the set of all the
N
Young tableaux. The symbol J>.l is the weight of the Young tableau, J>.J
= o:\o:~ .. .Q:~.
(4.5-l b)
j,( x ;w, l /a) = C;: 1(b;t)>.N( b;t,_l)'N-l-AN ...
Wil''-'' · 1,
N-l
C,
The first t wo formul ae (4.50a) a nd (4 .50b) indicat e t hat t he Hi-J ack polynom ials a re t he simu ltaneous eigenfunctions of t he first two conserved ope rators of the Caloge ro model wi t h
specifying their eigenvalue formul ae . The t hird property of t he Hi-Jack polynomia ls (4. 50c)
is called triangularity. The las t one is normalization. The fi rst three condit ions are based on
the observation in Chapter 3. Triangularity of t he Hi-J ack poly nomials with resp ec t to t he
dominance order plays an esse nt ial role in the uniq ue identifi cat ion of t he Hi-J ack po lynomi als.
Because of remaining degeneracy, a n eige nfunct ion cannot be uniquely id ent ifi ed on ly by the
eigem·alues for the first two out of the N commuting conserved operato rs. However , combining
the eigenva lues and t ria ngula rity, we ca n uniquely ident ify t he Hi-J ack polynomials.
P r o p os it io n 4 . 2 Let>. and J1. be distinct Young tableaux. If the two eigenvalues f or the Young
tableaux are equal,
(4.5 1)
= II Ck(>.I, >.2 , · · · , >.k+I; a),
(-1 .56)
k=l
where
Ck( >.I , >.2 , · · · , >.k+I; a) = (a),,_,k+, (2a + Ak-I - >.k),,_,k+, · · · (ka + >. 1 - >.k),,_,k+ , . (-1.5 7)
In t he above express ion , the sy mbol (f3)n is the Pochhammer symb ol, t hat is, (f3)n = (3(13 +
1) · · · ((3 + n- 1), (f3)o ~ 1. \\"hat we want to prove is summa ri zed as t he followin g propos it ion.
The symmetl'ic polynomials generated by the Rod1·igues formula (4.55) satisfy th e definition of the Hi-Jack symmet1·ic polynomials (4 .50}.
Prop o s iti o n 4 . 3
The first two out of four requ irements (-1. 50) are derived from t he followin g propos it ions.
Propos it io n 4. 4
(-1.58)
then we can not define the dominance 01·de1· between the two Young tableaux ). and JJ..
The prop osition asserts t hat we can distinguis h any two eigenfun ct ions wh ich share comm on
first t wo eige nvalu es E 1 and E2 by t riangula rity of the Hi-J ack poly nomia ls.
In ord er to wri te clown the Rodrigues formula for the Hi-J ac k polynom ials, it is com·enient
to introdu ce the foll owin g sy mbols (cf. eq. (4.18)):
jEJ
(dj, + ma)(d12 + (m + l )a) · · · (dj, + (m + k- 1)a ),
(4.55)
with t he const a nt C, gh·en by
= I: Ak ·
k=l
II aJ,
(-1.5-la)
Us ing th e raising operators (4.54), we can write clown the Rodrigues formula for th e Hi-J ack
polynomi a ls j,( x ;w, 1/a) as
~"''
or 1~1<1'1
w,, (a, l /2w) =
b;t
(4.50c)
D
o:jd 1,J, fork= 1, 2, · · ·, N- 1,
Js;{l,2, .. ·,N)
IJI=k
(4.50b)
E2 (>.)j,( x ; w, l /a),
I:
bt =
k= l
Prop os it io n 4. 5 The null opemt01·s ni+ I,J, which are defined by
nk+I,J
= do,J,
J ~ {1 , 2, · · · , N},
Ill= k + 1,
(-1.59)
satisf y
(4.52a)
(4.52b)
(-1 .60)
62
63
which completes the proof.
Proposition 4. 6
[I2 ,btJI
4.2. HI-JACK SY1\1METRIC POLYNOMIALS
={bt(21 1+k+ak(N-k))+
Sym
L
J~{l,2, .. ·,N}
9k+1,Jnk+1,J}I
.
(4.61)
Sym
IJI=k+l
Here 9k+l,J is an unspecified nonsingular operator that satisfies 9N+l,J
As byproducts of the proof of the last two requirements of Proposition 4. 3, we notice the
following results.
The expansion coefficients of the Hi-Jack polynomials C>.W>.~(a, 1/2w) are
polynomzals of a and 1/2w wzth znteger coefficients. This property is analogous to that stated
by the Macdonald-Stanley conjecture for the Jack polynomials {41, 50, 69}.
Proposition 4. 7
= 0.
The first requirement directly follows from Proposition 4. 4 . For a while , we forget about
the normalization constant. From the l.h.s. of eq. (4.50a), we get
Proposition 4 . 8
J>.(al, at·· ·,a~; 1/a) · 1 = J>.(x; w, 1/a).
I,(bt)A.V(b"j,_ ,)>-.v-1-AN ... wn>-1-A2 ·1
=([I,, (bt)>-"(bt_,)>-.v-1->.v ... (bi)>-1->.']
+ (bt)>-"(bt_,)>-v-1->.,,· . .. (bi)>.1->.'It)
(4.68)
·l.
(4.62)
We also notice the stronger form of the triangularity of the Hi-Jack polynomials (4 .50c). \\"e
define the "weak" dominance order among the Young tableaux by
Because of eq. (4.21d), the second term of the above equation Yanishes:
d
I 1· 1 =
L
a!ak · 1 = 0.
I
I
k=l
k=l
L J.l.k :<::: L >.k for alii.
J.1.9 ~
N
(4.63)
(4.69)
k=l
Namely, the definition of the dominance order (4.39) is given by adding another condition,
IJ..I = IJ.i.l to the above weak dominance order. The stronger form of the triangularity (4.50c) is
given by the following formula.
Then using Proposition 4. 4 , we get the expected result :
I,(bt)A.V(bt_,)>..v-1->..v .. . (bi)A1-A2 ·1
= (
N-l
2: k(>.k- >.k+,) + N>.N ) (bt)A"(bt-,))."_ 1-A" . .. wn>. 1->.,. 1
Proposition 4. 9
k=l
N
=
L
>.k(bW·v(bt,_,)A.V-1-A,V . .. wnA1-A2. l.
J>.(x;w, 1/a)
(4.6-l)
d
k=l
(4.70a)
and 1~1=1~1 (mod2)
Iz · j 0 (x;w, 1/a) = E2 (0)j0 (x; w, 1/a) = 0,
W>.>.(a) = l.
N-k
Then the following formula straightforwardly follows from the aboYe.
Proposition 4. 10
we ha,·e
N-k
l2bt(bt)>-'(bt_,)>., _1->., ... (bi)>- 1 ->., ·1
(4.70b)
(4.65)
by using eq. (4.21d) because j 0 (x;w , 1/a) is equal to 1 as a polynomial. By inductive assumption, eq. (4.50b) holds up to >. = {>. 1, >. 2, · · · , >.k, ....___,__...
0, · · ·, 0}. Then for >. = {>. 1 + 1, >. 2 +
+ 1,0....___,__...
, · · · ,0},
1 ) (1>-1-1~1)/2
W>.~(a)m~(x).
( 2w
~<A
The second requirement (4 .50b) is shown by induction. It is easy to show it for>.= 0,
1, · · · ,>.k
=
1 ) (1>.1-1~1)/2
W>.~(a)J~(x ; 1/a).
( w
J>.(x; w, 1/a) = J>.(x; w, 1/a) +
= ([~z,bt] +btJz)(bt)>.'(bL)>.'_ 1->., .. . (bi)>. 1->., · l
d
2
(4. 71)
1•$>. and 11•1<1>.1
and 1~1=1>.1 (mod2)
(4.66)
From the inductive assumption and Proposition 4. 6, we get
I2bt(bt)>.'(bt-t)>-•-1->.,. (bi)>.1->., ·1
k
= bt ( £2({>., ,,\z, .. · ,>.k.o, · .. , 0})
X
+ 2~>. 1 + k + ak(N- k))
(bt)>.'(bt_ 1 )>.,_ 1->.' · · · (bt)>. 1->. 1 ·1
= Ez( {>.,
+ 1, Az + 1, ... , ).k + 1, 0, ... '0} )bt(bt)A' (bt_,)>-,_ 1-A' ... (b"t)A 1-A 2 . 1t4.67)
During the discussion in Section 4.1, we have noticed that the Hi-Jack polynomials should
reduce to the Jack polynomials in the limit w-+ oo:
J>.(x;w
= oo, 1/a) = J>.(x; 1/a).
(4.72)
In Proposition 4. 10 , the abo,·e relation is expressed in more detail. Proposition 4. 8 giws another relationship between the Jack polynomials and the Hi-Jack polynomials. The relationship
64
between the eigenfunctions of the Hamiltonian He given by the QIS!Il (4 .16) or the eigenfunctions (4 .22a) and the Hi-Jack polynomials is now clear. Several bases for the ring of homogeneous symmetric polynomials are known [50, 69]. The power sums, the monomial symmetric
polynomials and the .Jack polynomials are examples of such bases. Thus, the transformation
between the Hi-Jack polynomials and two kinds of the eigenfunctions of the Hamiltonian He
is the transformation between the bases of homogeneous symmetric polynomials. Defining the
transformations by
ff
f_;"
1
1
= J; ,
({m;}) = J>.,
4.3. PROOFS
The explicit forms show the fact that the Hi-Jack polynomial is a one-parameter deformation
of the Jack polynomial ,
]>.(x; w = oo, 1/a) = J>.(x; 1/a).
This has been clarified as eq. (4. 72) by the discussion of the common algebraic structure of the
Calogero and the Sutherland models in Section 4.1. The above Hi-Jack poly nomials coincide
with the seven orthogonal symmetric polynomials (3.34) given in Chapter 3 up to normalization.
Orthogonality of the Hi-Jack polynomials follows from the following proposition.
(4.73a)
( {P>.})
(4.73b)
Proposition 4. 11 The Hi- Jack polynomials are the simultaneous eigenfunctions of all the
commutmg conserved operators Unln = 1, · · ·, N} with no degeneracy in the eigenvalues. Hence
they are the orthogonal symmetric polynomials with respect to the following inner product:
we have
ffj({<P>.}) = j). ,
(4.74a)
= i>.·
(4.74b)
f.r'J({cp;})
j'"" IT dxkl¢gi J>.(x;w, 1/a)j~(x;w, 1/a)
2
-oo k=l
ex: 6>.,w
Note that the transformation (4.73b) is nothing but the expansion of the Jack polynomials
in the monomial symmetric polynomials (4.38b). Propositions 4. 7, 4. 9 and 4. 10 and the
triangularity of the Hi-Jack polynomials (4.50c) are observed in the explicit forms. The explicit
forms of the first seven Hi-Jack polynomials are
j 0 (x; w, 1/a)
j 1 (x;w, 1/a)
j 1 ,(x;w, 1/a)
(a+ 1)j2(x; w, 1/a)
J0 (x; 1/a) = m0 (x) = 1,
J 1 (x ; 1/a) = m 1 (x),
a N(N -1)
J 1,(x; 1/a) +J0 (x; 1/a)
2w
2
a N(N -1)
mi'(x) +m 0 (x),
2w
2
1
1
(2a
+ 1)h,I (x; w, 1/a)
2w
(N-1)(N-2)
JI,(x; 1/a) +-a
·
JI (x; 1/a)
2w
2
1 (N- 1)(N- 2)
( )
711 13 ( X ) + -a
m1 X ,
2w
2
(2a + 1)h,I (x; 1/a)
(4.75c)
1
-W
(a 2 + 3a + 2)]J(x; w, 1/a)
(a
(a
-
+ 1)mi(x) ,
(4.75e)
In order to prove Proposition 4. 2, we need a preparation . We introduce an elementarv
deformation which will be applied to the Young tableaux later. Let x = {XI, · · ·, XN} be a se-t
of N nonnegative integers and X,v be the set of all x's, which includes Y,v, Y,v c X,v. Then
we can introduce a deformation R1 : X,v -; X,v for 1 :S I :S N- 1,
when X satisfies the condition
we notice
x1 > Xl+I·
Defining the dominance order in X,v by eq. (-1.39),
(-178)
This means that the elementary deformation R1 generates a smaller set of N nonnegati,·e
integers from a set of N nonnegative integers. \\'e want to show that successive applications of
tl;; elementary deformations to a Young tableau .\ generate all the Young tableaux J.l satisfying
(4.75£)
2
+ 3a + 2)JJ(x; 1/a)- ]_(a 2 N 2 + 3aN + 2)JI(x; 1/a)
2
+ 3a + 2)mJ(x) + 3a(a + 1)m 2,I(x) + 6a 2 mi'(x)
2w
3 ( hr2
w a l v +3aN+2)mi(x).
2
Unique Identificat ion of t he Hi-J ack P olyn omials
(4.77)
) 1 - a)(N- 1)(Na + 1)JI(x; 1/a)
2
(2a + 1)m2,1(x) + 6ami'(x)
-?(1- a)(N -1)(Na
Proofs
(4.75d)
1
-
4. 3
4 .3. 1
(a+ 1)J2 (x; 1/a)- -N(Na + 1)J0 (x; 1/a)
2w
+ 2ami'(x)- -N(Na + 1)m 0 (x) ,
In the next section, we shall prove Propositions 4. 2 - 4. 11 .
(4.75a)
(4 .75b)
1
(a+ 1)m2(x)
(4.76)
(4.75g)
J.L:S.\. Necessary properties of the elementary deformation are summarized in two lemmas,
Lemmas 4. 12 and 4. 13. The latter one "·ill be used in the proof of Proposition 4. 2.
The elementary deformation is not generally a deformation within the Young tableaux } ;\'
because a deformed tableau R 1(.\) does not belong to Y,v when the Young tableau .\satisfies
.\1 = .\I+I + 1. But we can show that successi,·e applications of the elementary deformations
gi'e us an deformation among the Young tableaux l';v.
---
--
-~
--
66
Lemma 4o 12 By iterative applications of the elementary deformations to a Young tableau
we can make A'
meet
E
A,
YN that is the largest in the dominance order of all the Young tableaux J1 that
D
11::;\
Jlk
= Ak,
for 1 ::; k ::; l- 1, Jll
= A1 -
1,
4.3. PROOFS
67
Thus following the discussions to find out the minimum suffices m and 11m for the Young tableau
A, "·e can find out mmunum sufficesm' and 11m• for the Young tableau (~.82). From the explicit
form (4.82b), \\e can 1dent1fy the m1mmum suffices as
(4.79)
m'
when A satisfies an inequality,
m-1
l,
~
(4.83a)
N
L Ak ::; (N- l + 1)(AI- 1).
R,.. o
; R,..m
R,..m 0 · ·
= R,..m
0
(4.80)
••
k=l
A proof of the abo,·e lemma is as follows. Since the Young tableau A satisfies the condition
(4.80), there is some suffix k ~ l + 1 such that Ak ::; A1- L We denote the minimum number
of such suffices by m + 1. This means that the Young tableau A E Y,v sat isfies
o R,(A)
i:(A)nm+l
R,(A)nm
· · · 0 R,(A)nm+l
0 · · · 0
o
0
0
(-L83b)
when m ~ l
+ 1.
Then we get
(U~a)
Thus we can apply Rm to A since Am> Am+l· Generally speak in g, when you can apply Rk to
a Young tableau J1, Rdt•) satisfies
+ 1 > Jlk+2 =
Rk(J.t)k+ 1 = Jlk+l
Rk(11)k+2,
for Jlk+l ~ Jlk+ 2. Therefore we can define a successi,·e operation of the elementary deformation,
Rk+l o Rk(Jl). By fur ther successive operations, we haYe
R,, o · · · R,+ 1 o R,(A), m::;
n::; N-
(H-!b)
1,
for the Young tableau A now we are dealing with. Due to the condition ( 4.80), there exists an
integer 11 such that
(4.81)
R,.. o o· oo R,(A) E YN, m::; 11 ::; N- 1.
=
0
0
•
= \v = A1
X= Rn,
o · · · R1 o
0
•
•
o R,..m o · · · o
R.n(A)
E }~v,
(H5)
where l ::; m ::; 11m ::; · · · ::; 111. By construction, we can readily sec that the Young tableau X
satisfies the condition (4 .79). The explicit form of the Young tableau is
If not , then the condition (4.81) is not satisfied for all 11, m::; 11::; N- 1. This means
Am+ 1
Iteration of the same procedure until m' becomes l finally yields a Young tableau,
- 1,
and the Young tableau A breaks t he condition (4.80) . We denote the minimum of such integers
by 11m. Then we get a Young tableau:
(4.82a)
Ak,
A1-l,
'; =!
1::; k::; l- 1
l::; k::; 111
N
:L
1'1- k=l
AI.,
k = 111 + 1
(-1.86)
k¢.nt+1
111
Ak,
+ 2::;
k::; N- 1
Let J1 be an arbitrary Young tableau satisfying the condition (~.79). \\oe shall compare the
partial sums of the two Young tableaux, X and Jl · From the condition (~.79), we see
(4.82b)
This Young tableau (4.82) satisfies the condition (4.80) because of the following relation:
N
. 0
R,(A)k
=L
k=l
Ak ::; (N- l
+ 1)(AI- 1).
m
m
k=l
k=l
L A~ = L Jl,
for I ::; m ::; l.
(487)
Due to the definition of the Young tableau and the explicit form of X (~086), we get the following
inequality:
~-~
--
--
68
m
m
k=l
for n 1 + 1 <::: m <::: N.
).ko
k=l
The above two relations are combined to give
m
L ).~ 2: L Jl,
for l
+ 1 <::: m
(4.88)
<::: N.
k=l
k=l
69
J ow we are in
of
the
h assumptiOn
fi
t e same rst two
The explicit form of X (4.86) also shows
L ).~ = L
4.3. PROOFS
D
position to pro,·e Proposition 4. 2. We use the reductiYe absurdit,·. From
the
·.
. proposition, eq. (4.51) , two different Young tableaux. ). fA'"E }- 'J\., gt\e
etgem·alues. Assume we can define the dominance order between the two
J
Young tableaux . Without loss of generality, we can assume ?t~A. Then from Lemma 4. 13,
there must be a way of applying iteratiYely the elementary deformations to the larger Young
tableau.). that ytelds the smaller Young tableau p, say Rr 0 . . . 0 R;(>.) = 11 . The iteratiYe
applicatiOnS cons1st of the elementary deformations (4. 77). Since the elementary deformation
does not change the wetght , the first eigenvalues E 1 for any X E XN and well-defined Rt(X) are
the same. Companng the second eigenvalues for X and R1(x) , we haYe
Thus eqs. (4.87) and (4.88) prove 11:'0X, which means that the Young tableau X is the largest
of all the Young tableaux satisfying the condition (4.79).
D
Text we show that we can make any Young tableau 11 such that 11:'0.\ by iterative applications
of the elementary deformations (4. 77) to a Young tableau >..
D
Lemma 4. 13 Let ). and 11 be two Young tableaux, >. , 11 E :l';v. If 11:'0.\, then we can obtain
the smaller Young tableau 11 by successive applications of the elementary deformations (4. 77)
,
E2(x)- E2(Rt(X))
= 2(xt- Xt+t
- 1)
+ 2a > 0,
because the inequality Xt > Xt+J holds to make R 1(x) well-defined. This inequality shows that
JteratJ\'e applicatwns of the elementary deformations Rk to X monotonically decrease the second
etgenvalu e E2 for the deforme~ X· Thus the second eigem·alues for any pairs of distinct Young
tableaux ). and 11 that meet tt:S.\ must satisfy the inequality,
E2(.\) > E2(Rro · .. oR;(>.))= E 2 (J1).
to the larger Young tableau .\.
We assume 11 is different from >., because the case 11 = ). is trivial. Then there is a minimum
number of l such that the 1-th partial sums of the two Young tableaux satisfy
I
I
k=l
k=l
2: >.k - 2: Ilk =
t:.1
> o,
where 6 1 is some positive integer. For conYenience, we add a superscript (l-1) to). to indicate
that the elements of two tableaux ).(I-I) and 11 are the same up to the (1-1)-th element. Since
D
This is contradictory to the assumption of the proposition, eq. (4.51) . Thus we have proved the
proposition. Proposition 4. 2 is essential to the unique identification of the Hi-Jack polynomials
just by diagonalizing the first two commuting conserYed operators. As a result, it will implicitly
play an important role to show that the Hi-Jack polynomials are the simultaneous eigenfunctions
of all the commuting conserwd operators.
4.3.2
Hamiltonian
We shall prO\·e Proposition 4. 4 . It is easy to prove the case k = N. From the definition
of 11 and bt, we have
the two Young tableaux satisfy ll:S>.(I-J), we have
N
l:[di,ala~ .. a~]
k=l
1-1
1=1
::: 1111-2: Ilk
N N
l:l:al···ai-t[di,aJ]al+ 1 ···a~
k=l
(4.90)
j=ll=l
Using eq. (4.2 l c), we get
k=l
<:::
(.V- l
+ 1)Jlt
= (N- l
+ 1)(.\1 -
6t).
The above inequality guarantees that we can apply the deformation shown in Lemma 4. 12 61
times to the larger Young tableau >.. The deformation yields a Young tableau ).Ul, which is th e
largest in the dominance order of all the Young tableaux that satisfy
D
v::;>., vk
= .\k,
1 <::: k <::: l- 1, v1 = .\ 1 - 61.
[11, bt]
N
=
N
L L
j=ll=l
(Nbt + aal · · · aL
Kj;)- a(1- 6jt)Kjt)a}+ 1 · ··a~
i=l
i;lj
N
La} (Kit- Kjt)a)+ 1 · · ·a~)
j,l=l
j#l
(4.89)
The Young tableau 11 also satisfies the co ndition (4.89). Thus we conclude that 11 ~;.Ul. This
means that we can apply the same procedure to ).U) again . Thus iteration of the procedure
finally yields the Young tableau ).(-"l that is equal to ?t.
N
al· ··al-taj(6jt(1 +a L
=
Nb~ ,
(H1)
which says the Yalidity of eq. (~.58) for the case k = N. 1\ote that we do not ha,·e to restrict
the operand in the abO\·e calculation.
70
For the case k
into two parts:
f N,
4.3
PROOFS
+a~(dh
we need more computation. First, we decompose the Hamiltonian [ 1
[r~,bt] =
L
JC{l
2 ... N)
-l;l~k·
[Ldi + Ld; ,a~d1,J]
iEJ
71
+a)··· (dj 1_, + (l- 1)a) ( (d1, +La) - (d; +La) )K;
x (dil+•
(4.92)
11
+ (L + 1)a) · · · (d1, + ka)}l
Sym
i~J
k
~~{ -aa~\{il)a)dl,J\{il)(d; + ka) + a~(dl,J- dl,J\{il)(d; + ka))
The first part of the r.h.s . of eq. (4.92) is calculated as
(4.95)
Substitution of eqs. (4.9-1) and (4.95) into eq. (4.92) yields
[L d;, a~d1,J]
[I~ ,btJI Sym
iEJ
k
LL{aJ, · ··aL[d;,aJ.JaL,···ai,dl,J
L
iEJ 1=1
+a~(dj, +a) · ·· (di•-• + (L- 1)a) [d; , dj, +La] (di•+• + (L + 1)a) · · · (d1, + ka)}
{ kaJdl,J
J<;;{l ,2,---,N)
[J[;k
k
k
L L {a}, · ·· a}._, (6;j,o},(1 +a L I<ili +a L Kid)- o)(l- cl;j1 )aKij,)oL, · · · ai.dl,J
iEJ I; I
jEJ
+aLL{(ajl'fj /;1
;~J
jfj,
+ (l + 1)a) · · · (d1, + ka) }·
{4.93)
We move exchange operators to the rightmost and utilize the restriction I
. Using eqs. (4.19)
Svm
and (4.53), we get
a~\Uda!Jdl,J\{il)(d; + ka) + a~(d 1 ,J- d 1,J\{il)(d; + kaJ)}}I
kbtl Sym .
+a~(dh +a)··· (di•-• + (l- 1)a)((dj, + la)- (d; + la))K;j1
(di•+•
}ISym.
Sym
(4.96)
Thus we have proved Proposition 4. 4 .
4 .3.3
Null Op erators
We shall prove Proposition 4. 5 . Since the function (bt)>.•(b[_ 1)>.,_,->., . -. (bi)>.,->.,. 1 is
a symmetric function of .V Yariables {x 1, x2 , · · ·, XN }, it is sufficient to prove the case J =
{1 , 2, · · ·, k + 1}. For breYity, we use the symbol nk+1 = nk+l,{l,- .. ,k+ l) hereafter. Then the
expression to be prO\·ed is
·
[L
d; , o~d 1 ,JJI
iEJ
Sym
k
kaJdl,JI
+a L
L a~dl,J\{il)(d; + ka)l
if/:J l=l
Sym
k
+aLL { ( o~\U}o}, l=i iEJ
i¥-jl
(-1.97)
Sym
Th is fo llows from
a~\Udal)dl,J + aH d1,J\{i) (dj, + ka) -
d 1,J\Ud (d;
+ ka))} I
k
kaJdl,JI
Sym
+aLLa~dl,J\{il)(d;+ka)l
i'/:J 1=1
[ni+l,btJI
Sym
.
(4.9-l)
Sym
~nk+ll Sym ,
fori<':k,
(-1.98)
where t he symbol ~ means that the term on the r. h.s. can be mu ltip lied on the left by some
nonsingular operator 0 , 0 · 0 = 0. We can easily wrify
nk+l(bt)!.'(bt_l)!.·-·-!.· ... wn>.,->., ·1 ~ nk+l(bt_l)>.,_,->., ... wn>.,->., ·1
Sym
The second part of the r.h.s. of eq. (4.92) is calcu lated as
[L
d;, o~d1,JJI
irf.:J
Sym
(-1.99)
k
using eqs. (-1.21d) and (4.98). For com·enience of explanation, we introduce a symbol [m] for
LL{aJ, --·aL,[d; ,o},]aL, - -- a}.d 1,J
iftJ l=!
+a~(dil +a)··· (dj,_, + (L-
a set {1 , 2, .. · ,m} with an integer m. We also introduce IJ
1)a) [d;, dj, +La] (di,+•
+ (L + 1)a) · · · (d 1, + ka)} I
Sym
with a set of integers J that
Sym
ind icates the operand is a symmetric function of xi, j E J. From the identity (4.53), we ha1·e
[k+lj
nk+ 1
lSym
=
nk(dk+ 1 + ka)
l[k+ll
Sym
72
4.3. PROOFS
k
I:
I:
a~.a~dvdk-l+l,J
J'!:;[k+!J Jl:;{k+2,-··,N)
[J'[=k-1
[Jj:/
I:
1=0
k
I:
I:
1:0 Jl:;{k+2.-··,N)
[Jj:/
a~bL,,k+ljdk-l+l,J·
Thus we have the following formula.
Lemma 4.15
k
{ dtKt k+l +a+ a
L Ki k+l
k
}
nk
j:::; 2
llk+!J
I:
a~bt-l,[k+tJdk-l+l,J,
/:0 Jl:;{k+2,-··,N)
[Jj:/
bt =I:
,
(4.100)
Sym
with bci, 1 = 1 and when
which means
lk+!J
nk+I
lSym
llk+lJ
,...,_, nk
(4.101)
.
Sym
,btJI
= 0, a~ = 1 and dk+l,J = 1.
We shall a lso use the following formulae:
Lemma 4. 16
Thanks to the above relation, eq. (4.98) fori> k follows from that fori= k, i.e.,
[nk+ 1
/11
[d,,a~J
~ nk+tl Sym ·
Sym
(4 .102)
-aal
L a~\{j)](iJ,
jEJ
[d,,a~J
aHl+a
We shall prove a stronger statement, namely,
t l[k+lj
[nk+l,ad
Proposition 4. 14
[Mj b+ IAIIJ/!NI
[nk+I' k
Sym
~ nk+l
[Mj /INI
Sym
Sym
,
J\I
2 N.
(4.105)
(4.103)
i
rf.
J,
(4.106a)
I: K,J), i E J,
jE[AIJ\J
(4.106b)
-a( alKt2K23 · · · Kkk+tKk+ll + · · · + al+t](k+ll)nk llk+lJ
Sym
nk
llk+ll
, l
rf. [k +
1].
(4.106c)
Sym
Here , the superscript [J\I] over Dunk! operators indicates that they are made from the Dunk!
operators (4.18) that depend not only on the variabl es x 1 , x 2 , • · · , X,y but a lso on XN+l, · • ·, XAf.
11
Note that n~~~~ and bt 1·' are symmetric under SN but not under SM. 'vVe just changed the
number of mriables of Dunkl operators (4.18) but do not change the definition of operators
made from them , such as conserved operators (4.24) and raising operators (4.52) and (4.54).
Namely, the indices and subsets in the summand of their definitions arc included in the set [N].
All the Dunk! operators in the remainder of this Section 4.3.3 will aill"ays depend on
x 1 , · · ·, XAf. \Ve shall omit the superscript [.i\I] in the following.
To pro,·e Proposition 4. 1~ , we need seYerallemmas. \\'e define the restricted raising operator by
bt,,
= L
a~.dt,J',
(4. 10-1)
The first two formulae can be checked from the definition of d, and a~ and commutation
relations (4.2 1):
IT aJ. [a,, aj]
alI:
jEJ j'EJ\{j}
-aal
L a~\{j)J(ii•
jEJ
[ala, , al
i"l=k
bt
'!Jf
L
J!:;[NJ
IJI=k
a~dt,J
rf.
J,
aj]
jEJ\{i}
a![ a,,
al]o~\{i}
+a![a!, a~\(i)J
AI
fCJ
where J is a set of integers. From the definition of the raising operator (4.5-1), we can easily
verify
rr
i
al(1 +a
L K,k)a~\{i)- aa) L
jEJ\(i)
k=l
k>'i
a~+ aal
L
a~\{k}K,k + aa~
kEJ\{i)
a~(1+a
L
jE[.I/j\J
a~\{j)J(iJ
L
kE[AIJ\J
J(ii) , iEJ
K,k- aal
L
JEJ\{•)
a~\{J)J(, 1
74
The last formula (4.106c) is proved by induction on k. By a straightforward calculation using
the commutation relations (4.21) and the definition of the null operator (4.59), we can confirm
the case k = 1:
[n2 , al]
121
1Sym
4.3. PROOFS
T5
~
nk/
Sym
, l ¢ [k + 1],
which proves the formula (4 .106c).
In the following, we often need to deal with the terms that do not have al appearing as an
expltctt factor on the left of the terms. When 0 represents a Dunk! operator of our interest
such terms are denoted by
.
,
oj
a,1. . . o
Lemma 4o 17 For i\1 2: n 2: k
+ 1,
we have
(.J.107)
Note that we have used the relation K 12
)2]
1121
=1
1Sym
Sym
in the above calculation. Assuming the
([n2, a~ .. · a~]+ a~·
validity of eq. (4.106c) up to the case k- 1, we can calculate the case k :
Jk+lJ
[nk+l• a!]
ISym
This formula is also proved by induction on k. The case k = 1 is verified as follows. From the
definitions of the operators and the first two formulae of Lemma 4. 16 , we ha1·e
·a~ [d 1, d2 +a]) I 1
o 1 -o
jk]
t ) IJk+1]
( [nk,al] lSym (dk+t+ka)+nk[dk+l,a 1] Sym
([d1, a~·
· · a~] (d2 +a)+ dt[d2, a~ · ·a~] + aa~ · · · a~(d 2 - dt)I<12 ) I
-a( (alK12K23 · · · Kkl + · · · + a!Kkl)nk-1(dk+1 + ka)
( d 1 a~
a~(l +a
+[nk, a!+t]
jk]
) ljk+l]
Kk+11 + al+ 1nkf(k+11
lSym
Sym
0
•
•
0
L
jE[M]\{2, .. ·,n}
a~- .. a~(d1(1+aK2 1 +a-£
=
-a( a!K12]{23 · · · Kk k+t + · · · + alKk k+1 )Kk+llnk-1
a~---a~(l+aK12+a
Using the merit of the restriction
, l ¢ [k + 1].
L
I<2i)d 11
0
0
•
a!,(d2 -
d1)K12) I
we have
( [nk+t, a~··· a!,]
ol-o
a,1. . . o
o!-o
~d 1 .
+a~· · · a~ [d1, d(2,- ,k+1J])
I t-o
a
([nk.a1 .. a!,]+a~ .. · a~[d1,dt,{2, 0 k)])(dk+1+ka)/
lSym
+(nk[dk+t,a1o··a~] +a~··oa~d1,{2o
-a((alK12K23 · · · Kk1 + · · · + alKkl)nk-1(dk+1 + ka)Kkk+1
1 1
-a( al K12Kn · · · Kk k+1 + · · · + alJ<k k+1) Kk+tlnk-1 + al+ 1Kk+11nk) / k+ ]
,kl[d1,dk+t] )j
o 11-o
o 11 -o
1) (nk + ank-1)
-a( a!J<12K23 · · · Kk k+1J(k+11 + · · · + alJ<k k+1 Kk+1 1)nk- 1 + al+ 1Kk+1 1nk) l[k+l]
[k+1]
lSym
.
(-1108)
Using the second formula of Lemma 4. 16 and eq. (4.21), 11·e can readily calculate the second
term in the r.h .s. of the abo1·e expression:
Sym
-a( (c,J K12K2J · · · f(k k+1f(k+11 + · · · + alKk k+1 f(k+1
-a( alK12K23 · · · Kk k+1J(k+11 + · · · + al+1Kk+11)nk
o!-o
As the inductive assumption , we assume that the lemma is valid up to k- 1. Then the case k
is expressed as
[k+l]
[nk+1, al]
I<2i))/
M
j=n+l
[k+l]
lSym
[k+lJ
l[k+l]
f{k k+ 1
= 1
,
lSym
Sym
+ aa~
r=n+I
-a( (alK12K23 · · · Kkl + · · · + alKkl)nk-1(dk+1 + ka)
+a!+t Kk+ 11nk )
K 2i)
Sym
AI
=
((a~- a~nk+[nk.a~ .. ·a!,])(l+aK1k+t+a L
j=n+I
Kk+1i )
76
+aa~ · · · a~d 1 ,(2,. 'kj(dk+1- dl)K, k+l) Io,-o
1
(([nk.a~·· · a~] +a~···a~[d 1 ,d 1 ,(2,
,k)]
+a~···a!,dt,{2,
4.3. PROOFS
77
Equations (4.110) and (4.111) prove Lemma 4.17.
As a step toward proving Proposition 4. 14 , we prove the case N
Proposition 4. 18
,k)d, )
=k +
1.
M
x(1+aKlk+t+a
(4.112)
L
Kk+li)
j=n+l
+aa~. ··a!, (d 1,( 2,. .. ,k}K1 k+ldl (([nk. a~ ... a!,]
Both sides of the above equation are symmetric under permutations of indices 1, . .. , k + 1.
From the first two equations of Lemma 4. 16 , we have
d1,(2,. ·,k)d,K, k+l)) I
o,1 . . . o
.
AI
+a~··· a~ [d1, d1,(2,-··,k)]) ( 1 + aK, k+l +a . L Kk+l i)
[nk+l• at,··
k+l
i
= IT aj.
Commutators among the operators made of d1 operators are also
j=l
j'l'i
Substitution of the above equation into eq. (4.108) yields
operators made of d, and Klm with 1 ::; I, m ::; k + 1. Thus we can say that the raising operator
b([k+IJ and the l.h .s. of eq. (4.112) have factors al · · .0 . · · al+l on the left. Therefore, in order
to prove Proposition 4. 18 , it is sufficient for us to prove the following expression:
,k+l)J)Iol-o
a~ ... a~] +a~·· · a~ [d 1, d 1,( 2,. .. ,k}]) (dk+1
AI
L
+ ka + 1 + aK, k+l +a
J(k+l i)
[nk+l, b([k+IJ] [k+l'l
;=n+l
lSym
(4110)
or -a
~ nk+l l[k+IJI
Sym
121
a! (1
+a~ K 1j) + aH1 +a~ K2j)
AI
AI
M
j=J
"£
L
1[2J
I
1
Sym o 1 -o
or-o
)
n2
1[2]
Sym
Iol-o
~nz.
We assume that eq. (4.114) is ,·alid up to k -1. From the definitions of the null operator (4.59)
and the restricted raising operator (4.104), we can decompose them as
Kk+l i) I ,_
;=n+l
M
d 1 (dk+ 1 +ka+1+aKlk+l+a
(-1.114)
Sym
aH1+a:LK2i)nz
a~·.· a!,] +a~·· · a!, [d1, d1,(2,..,q]) (dk+l + ka + 1 + aK1 k+l +a
.
[dl(d2+a),a!(d,+a)+a~(d2+a)JI I
(
This formula can be readily verified by the inductive assumption:
or -a
This is proved by induction on k. The case k = 1 can be verified by a straightforward calculation:
The second term of the r.h.s. of the above expression has d 1 at its rightmost . Thus the proof
of eq. (4.107) reduces to the proof of the following relation for the first term of the r.h.s. of eq.
(4110):
([nk.
(4.113)
where O,i is some unspecified operator that can be written by d1 and K 1m with 1 ::; I, m ::; k + 1
and al · · .V · · · al+l
(([nk.
.
j=l
;=n+l
(4.109)
([nk+ 1 ,a~···a~] +a~···aqd,,d{2,
N
.0. ··atk+l ]'\' ~t . . .~ ... L.(k+lviJJ
~t /"
- ~"""1
o, 0
b([k+ll
=
albt_ 1,( 2,. .. ,k+l)(d, + ka)
+a~··· al+ 1d1,{2,-··,k+l}•
nk+ 1 = dldl,{2,···,k+l)·
f(k +li )
Then the l.h.s. of eq. (4.114) of the case k is cast into
j=n+l
M
(dk+ 1 +ka+ 1 +aKlk+l +a
L
Kk+li)d,
+ [d, ,dk+l +aKlk+d
j=n+l
M
(dk+l
+ ka + 1 +a!(, k+l +a. L Kk+l i)d,
ntb+k-1,{2, .. ·,k+l) (d I + ka) +at2 ... akt+ldl,(2, .. •k+l)
[d I d 1,{2,-.. ,k+l)·~l
ll
J=ll+l
+a(dk+l- d,)f{, k+l + a(d,- dk+l)K, k+l
AI
(dk+l + ka+ 1 +a!(, k+l +a
L
j=n+l
Kk+li)d, ~ d 1.
,k+IJI
Sym
t
o 1-o
(d1 [dl,{2, .. ·.k+l}•al]bt_,,{2,-·,k+l)(dl + ka)
+ ( [nk+l,
a~.· · ak+l] +a~· · · al+d d
1,
dt,{2,- ,k+l) ])c11,{2, ,k+l))
~~ky:.ll-o · (4.115)
78
From Lemma 4. 17 , we can east·1 y con fi rm th a t the second term of the r.h.s. of the abo,·e
equation satisfies the following relation:
4.3. PROOFS
79
which complete the proof of Lemma 4. 19. Using Lemma 4. 19 , we can calculate the l.h.s. of
eq. (-1.116) as
~
-ad, ( a~/(23 · · · Kk+l 1 + · · · + al+l Kk+t t )dl,{2, .. ,k}bt_ 1,{ 2, ··,k+t) (d 1 + ka)l[k+tlj
.
Sym at-o
[k+tll
= nk+t lSym
·
a! -o
(4.117)
Paying attention to the relations,
Thus we ha,·e to show the following relation so as to prow eq. ( 4.114):
[k+tll
dt [dt,{2,-··,k+l}• al]bt_l,{2,-··,k+l}(d, + ka) lSym ol-o
~
l[k+l[l
nk+t
Sym
ol-o.
(4.116)
We need a formula that is sim ilar to the third formula of Lemma 4. 16 :
[d,,a11]1
=0,
a t -o
dk+tdt,{2,··,k}=nk,{2,···,k+t),
1
nk{2
k }(d l +ka)-n
' ···,·+l
- k+l•
we can furth er calculate eq . (4.117):
Lemma 4. 19
{2 ...
[d 1,{ 2,. .. ,k+l}• al] Sy~·
1
k+l}
t
)
1{2,-··,k+l}
= -a(a~/(23 · · · I<k+tt + · · · + ak+tKk+tt dt,{2, ··,k) Sym
-ad, ( a~K23 · · · Kk+lt + · · · + al+t](k+t 1)dt,{2, .. ,k}bt_ 1,{ 2,. .. ,k+l} (d 1 + ka)l[k+tll
Sym a!-o
-a(a~K23 · · · Kk+tt
+ · · · + al+tf(k+tt)dk+tdl,{ 2,. .. ,k)bt_ 1,{ 2,. .. ,k+t}(d 1 + ka)l[k+IJI
Sym a!-o
This lemma is also proved by induction. Verification of the case k = 1 is eas ily done:
[d2 +a, al]
=
-aa~Kt2·
( [nk,{2, ··,k+t), bt_t ,{2, ··,k+t}] + bt_t ,{2,-··,k+t} nk,{2, ··,k+t}) (d1 + ka)l [k+tll
Sym
'vVe assume the lemma holds up to the case k - l. Using the inductive assumption, we can
calculate the l.h.s. of the lemma of the case k as
tll{2,-··,k+l}
[dl,{2,-··,k+l}• a, Sym
( [nk,{2,. ,k+l}, bt-1 ,{2, ··,k+t}] (d, + ka) + bt_t,{2,-··,k+l)nk+t) Ilk+ til
.
Sym o!-o
Note that the second term in the last expression is the expected null operator nk+t· Due to the
indu ct ive assumption, the first term can be cast into the followin g form:
t ) 1{2,-··,k+l}
([dl,{2,. .. ,k}• a!] (dk+t. + ka) + dt,{2, ··,k} [dk+t, a,] Sym
[nk,{2,-··,k+t). bt-,,{ 2,. ·,k+t}] (d1 + ka)
(-a( aV(23 · · · Kkl + · · · + alKkt )dt,{2, ··,k-l) (dk+l + ka)Kk k+t
{2,-··,k+l}
-adt ,{2,- ·,k}ak+t Kk+l 1)
1Sym
[k+ljl
lSym of-o
~ nk,{2,-··,k+l} (d,
+ ka)
l[k+ljl
Sym ai-o
= nk+t l[k+ljl
.
Sym a!-o
Thus we ha,·e praYed eq. (4.114) and hence Proposition 4. 18.
We need a few more formul ae to pro,·e Proposition 4. 14 .
-a( ( a~K23 · · · Kk k+tKk+t 1 + · · · + alKk k+IJ(k+l 1) (dl,{2,· ·,k} + adl,{2,-··,k-l})
{2,-··,k+l}
+al+tf(k+ltdl,{2,-··,k} + [dl,{2,-··,k} •ak+ 1]Kk+lt )
o 11 -o
Lemma 4. 20 For all sets of positive integers J = {j 1, ]z, · · ·, j 1} ~ [N] such that J
{1 , 2, .. ·, k + 1} = 0, we have
n
1Sym
-a( (a~/(23 · · · Kk+tt + · · · + ak+t[{k+t 1)d1,{2,·,k}
(4.118a)
+a( a~K23 · · · Kk k+tKk+l t + · · · + alKk k+l f(k+l 1)dl,{2,-··,k- l}
{2,-··,k+l}
-a( a~K23 · · · Kk k+l + · · · + al Kk k+t )dt,{2,-· ,k-l)J(k+l 1)
1
Sym
-a(a~/(23 · · · Kk+l 1 + · · · + ak+tKk+tt)dl,{2, ··,k)
{2,-··,k+l}
1Sym
(4.118b)
(4.118c)
80
These formulae can be straightforwardly verified by using the
equation of Lemma 4. 16 and the fundamental commutation
(4 .21). For convenience of the proof of the above formulae, we
an unspecified non-singular-operator-,·alued coefficient. This
the expressions in the foll ow ing way:
definitions of operators, the third
relations among Dunk! operators
introduce a symbol "' that means
symbol enables us to compactify
4.3. PROOFS
81
. )
(
l[k+JjU(o)
x ( dJ+J +)a
· · · dk+J + ka)
k
Sym
"'\' a(K - 1)
-::]-1
'1
nk+J
~
nk+J
l[k+!Ju(i)
Sym
~
nk+J
l[k+![u(i)
Sym
' i
d
'F
[k + 1).
By iterative use of the above formula, we have
(some invoh·ed expression)~ (an operator)¢* (some involved expression)= "' (an operator),
(some invoh·ed expression ) x (an operator)+ "' (an operator) = "' (an operator).
( nk+J (dit + (k + 1- l)a) ... (d Jt + ka) _ dk+I-[,Jilk+l ) /WI
Using this •-symbol, we shall prove Lemma 4. 20 . By iterated use of eq. (4.101) and the third
formula of Lemma 4. 16 , we have
([nk+l· (dit + (k + 1 -lla)] + (dit + (k + 1 -lJa)nk+lrm
x(dh + (k + 2 -l)a) · · · (dJ, + ka)/[N( + "' nk+J/[Nl
Sym
Sym
...
(
)
I[NJ
[NJ
.,. nk+l dh+(k+2-l)a ···(dj,+ka)
+ "' nk+J/
Sym
=
Sym
"'nk+ I/ [N( .
Sym
Thus we have proved Lemma 4. 20 .
Now we are ready to prO\·e Proposition 4. 14 . We shall pro,·e it bv induction on k Fi st
we shall check the case k -- 1· From th e d e fi mtwns
..
..
·
· and
r '
of the raJsmg
operators
eqs. (4.5.J)
'
(4.104), we ha,·e
which proves eq. (4.118a). Verification of the second formula is simple. From the definitions of
the operators nk+J-I and dk+J-I,h we have
[k+Jj,[N[
+]/[Nj
[
I[NJ
[n2,bJ Sym = n2,bi,[2J] Sym +
[Nj
~[n2 ,a)(d; + aJ]/Sym.
N
to Proposition 4. 18 ' tl1e fi rs t t erm 0 f t he r ..
h s. of the above equation is confirmed
According
to
be
[n[
l
nk+l-ldk+J-I,J Sym Sym = d1(d2+a) ··· (dk+J-I+(k-l)a)(dj, +(k-l+1)a) . .. (dj, +ka)/
Sym
.
We denote a sequence of coordinate exchange operators that transform the indices {1 ... , kl + 1,jJ , · · · ,]t} to {1, · · ·, k + 1} by K~ . Then we have
'
[n2, bi, 121 ]
[Nj
I[Nj
~ n2 .
ISym
Sym
(.J.120)
The second term is calculated as
t
l[[NJ
[ n2,a;(d;+a)
~
=[n2,a1](d;+a)
I[N[
~
[!VI
+a1[n 2,(d;+aJ][".
[Nj
a1[n2,(d;+aJ]
I
Sym
which prO\·es eq. (.J.ll8b) The fo llowing commutation relation is convenient for the proof of
eq. (4.118c):
11k+ll
~n 2 [
(.J.l21)
~
From the third formula of Lemma 4. 20, we notice that the second term
(4.121) satisfies
[ ll
(4.119)
~f the
r.h.s. of eq.
[N(
.
(.J.l22)
Sym
Using eq. (.J.l06c), we can verify that the first term of the r.h.s. of eq. (.J.l21) reduces to
[k+!JU(i}
dj
=
Sym
(.J.l23)
(4.124)
Thus we have confirmed that the proposition holds for the case k = 1.
By inductive assumption, the proposition holds up to k-1. What we like to show is that the
commutator between nk+I and each term of the decomposition of the raising operator (4.105)
is similar to nk+I, i.e.,
(4.125)
[nk+t•
83
lk+l]
llk+Il
~ nk-I+I
.
lSym
Sym
Then the second term of the r.h.s. of eq. (4.129) is calculated as
[nk-I+I• bt_l,lk+IJ]
[nk+I-1, bt-l,lk+l]]
lk+ll
IINl
dk-I+I,J
lSym
Sym
~
INl
nk-I+Idk-I+I,J
ISym
INl
nk+I
.
(4.132)
ISym
On the other hand , the second term of the r.h.s. of eq. (4.128) is separated into two parts as
t [
b+
IINl
t
IINl
IINl
a, nk+I• k-l,lk+Il dk-I+I,J s m = a,nk+Ibt_l,lk+Ildk-I+I,J
- a~bt-l,lk+Ilnk+tdk-I+I,J
.
INl
Y
ISym
a~bt-l,lk+tl]dk-l+t,JIINl + a~bt-l,lk+tl [nk+t• dk-l+t,J J/INl .
~m
(4.131)
l
Using the Leibniz rule, we decompose the l.h.s. into two parts:
=
PROOFS
From the inducti,·e hypothesis, we ha,·e
Summarizing the results, we have
[nk+t, a~bt_l,lk+l]dk-l+t,J]
4.3
(4.126)
~m
Sym
Sym
.
.
(4.133)
In an analogous way to the venficat10n of the first and the second terms in the r.h.s. of eq.
(4.129): we can confirm both the second and the first terms in the r.h.s. of eq. (4.133) are
respectively stmtlar to nk+I· Then we obtain
Then from the third formula of Lemma 4. 20 , we notice that the second term is similar to nk+I:
+
IINl
a,t [ nk+I, bk-l,lk+tl]dk-I+I,J
~ nk+I IINl .
(4.127)
Sym
Sym
[Nl
INl
(-Ll3.J)
Equations (4.130), (4.132) and (4.134) yield
Our remaining task is to check the first term. When l 'it IJI = 0, the first term is similar to
nk+ 1 because of Proposition 4. 18 . When l # 0, we ha,·e to do some calculation:
[nk+I•
a~bt_l,lk+IJ]dk-I+I,J Sym ~ nk+IISym .
I
(4.135)
Thus we ha,·e proYed Proposition 4. 1-! and hence Proposition 4. 5 .
(4.128)
From the first formu la of Lemma 4. 20, the first term of the r.h.s. of eq. (4.128) is calcu lated
as
INl
[nk+I, aj] bt-l.lk+Ildk-I+I,J
ISym
[nk +I,
a,]t llk+ll bt_l,lk+l]dk-I+I,J IINl
Sym
4.3 .4
R a ising Op erators
Proposition 4. 6 is given in a form whose operators depend on N variables x 1 , ... , x,v. It is
convenient for us to explicitly indicate the number of ,·ariables, for we shall change the number
of variables during the induction procedure.
Propos itio n 4. 2 1
+[NlJI[Nl
[I2INl( N ) , bk,IJ\'l
Sym
+[Nl ( 2/ INl (N)
= { bk,[N[
1
+ k + ak(N- k) ) +
Sym
INl
nk+I-Ibt-l,lk+ l]dk-I+I,J
bt_1,lk+tlnk+I-Idk-l+t,J
ISym
INl
ISym
L
J<;;[Nl
IJI=k+t
[Nl
INl
9k+I,Jnk+I,J
} /INl
, N ~ k.
Sym
(4. 136)
H er·e
INl
+ [nk+I-1· bt_ 1,lk+tl] dk-l+t,,, Sym .
(4.129)
Note that the operators bt_l,lk+tl and dk-I+I,J are im·ariant under the permutat ions of indices
1, · · · , k + 1. Using the second formula of Lemma 4. 20, we can verify that the first term of the
r.h.s. of eq. (4 .129) reduces to
INl
bt_l,lk+tlnk+t-ldk-l+t,J
INl
~ nk+tl Sym .
ISym
(4.130)
g~'~1I,J
is an unspecified nonsingular operator that satisfies g!:li,J
= 0.
Note that 11·e haYe introdu ced the notation
"'
I);'tl(N) = L(d\''11)".
(4.137)
i=l
It is obvious that Proposition 4. 6 is tantamount to the above proposition. We shall prO\·e
Proposition 4. 21 by indu ction on k and !V. Precisely speaking, we usc the induction on l,
which relates k and N by k = l + 1 and N = l + J\I with arbitrary integer .\f. The plan requires
several lemmas.
84
4.3. PROOFS
85
Corollary 4. 23 For 1 ::; i < N ::; M, we have
Lemma 4. 22
al[MJ .. .~ ... a~Ml((l+aK; 1
M
+a
t[MJ t [MJJI
[I!MJ(N _ 1), at1[MJ ... ~ ... aN-laM
M
L
L
K 1d+(1+al(; 1 +a
k=N+l
Kjk)djMI
oi-o
=atM [MJ[IIM-i](N1) ,alt[M-1] ... ~ ... u.N-1
~t[M-l]ll
2
k=N+l
AI
L
+d}"'l(1+al(;1 +a
K1k)), 1::0:i,j::O:N, ifj,
I
ot-o
.
(4.139)
O{M-IJ ...... o[M]
(4.138a)
k=N+l
[(d}Ail)2, ali"'i .. .~ ... a~uJJI '
o 1 -0
.
-aa2t[M] · · ·~ ·· · aNt[M] a 1tfMJ ( d1[AI] K1j + K,j d[AIJ
i
+I( 11 +a }( lj (J( li +
j
= i or N + 1 ::; j
::; !If.
M
"\'
L.. }( lk )) ,
k=N+l
(4.138b)
Both formulae are readily verified from the first two formulae of Lemma 4. 16 . Using the second
formula of Lemma 4. 16, we can prove eq. (4.138a) as follo\\'s:
[(drfJ) 2, alfAIJ · · .~. · · aWfJ]
[AI] .. .~
~1
d)[M]~t
i
t[M] (1
··aN
· a~A/]( 1 +
+al[AIJ · · .V
"\'
L.. Kjk)
k=N+l
atfAil · ·
v ··
oi-o
~
j=l
= [(d)Ml)2, al[MJ ... ~ ... a;~':_llaL[MJll
ai . . . o
In a similar ll'ay, the r.h.s. of eq. (4.139) is calculated as
atM JMJ[riM-!J(N1) ,0:1t[AI-lJ
2
N
Af
L
a~M 1 ((l + aK;i +a
N-1
= [2:(dY!JJ2,a:rMJ ..
K 1k)djM I
k=N+l
i
t[AIJ t [AI]ll
. · QN-lQM
k=N
M
L
aK;1 +a
[/2[M](JV- 1), at,[AI] ... ~.
= -aa1[Mj. · · atN-1 [M] aM
t [MJ(iMJy
f" d[M j f'
(
~~~
)
;
''; + ,,; ; + ,,; + ai<li K 1; + L.. K 1k) .(4.140)
M
+ aK;i +a
The restriction I .\1_ _ M mea ns th a t we respectively
·
.
. the Dunkl operators a)·[If] , a)·11 1
1dent1fy
11 01 1
01
and d[MJ with atiM-l] [M-1]
[M-lJ
.
'
'a;
and d;
, where ~ ::0: !If - 1. The l.h.s. of the ab01·e formula is
' d
ca Icu Iate as
L
Kid+ (1 + aK;i +a
k=N+l
Kjk)djAII
k=N+l
_
t [AI](
-aM
tfAI-lj
-aa2
~
t[At-lJJI
... . ··QN-1
t
... aN-I
I
at . . . a 0[.\t-!J ....... oPIJ
[Al-l]
N
L
+d}M 1(1+aKij+a
k=N+I
K 1k)) , 1::0:i,j::O:N, ifj.
Verification of the second formula (4.138b) is don e with the help of eqs. (4.106a) and (4.106b):
[(d~'li)2, atfMl ... ~ . . . a~uJJI
a
t-o
ll'hich coincides ll'ith eq. (4.140). Thus 11·e haYe confirmed Corollarv 4. 23.
The folloll'ing formulae are Yalid for arbitrary number of Yariabies.
-aiMla2t[MJ
... ~ ... atN[AI]atK,... .~ ... aN
t[AIJ aJtJ( l;·d[MJ)
J
J J aat[MJ
2
J I
ai-o
i
v
.
t[M] t(d[AI]}' . }' d[AI]
aN a 1 i '' 1 +
1
''i
+
(
1 +a
"\'
'
' )
J, 1k)l1 1j
L..
Lemma 4. 24 For· M:;:,: N, we have
kE([MJ\([Nju{j} ))U{ l }u{i}
-aa2t[Mj .. ) . . . aN
t[Mj ait (dfMJJ(
1
ij +
j = i or N
+ 1 ::;
J(
d[Mj + }( 11 + aJ,lj(I,,;
'
' +
iii
~
' )
L.. J,lk)
k;:;N+I
j ::; i\1.
[I):' 1i(N -1),d!~ 1 1]
= a((!V
-1)(d!~ 1)"- !),' 1i(N
1
I[N]
-l)) Sym ,
[Nj
d[.IIJ·ll'
[( d[M])"
N
' l,[N]
Sym
Thus Lemma 4. 22 is praYed. Lemma 4. 22 leads to the following formula.
[N]
ISym
(nna)
[J\'j
= ad[AIJ
(JfAI](tV
1)(d[AI])"
)I
J,[N-l)
n
· -1)- (N·
1\
,
Sym
1 >? (4.142b)
"'J\
v
_ -·
CHAPTER .J. ORTHOGONAL BASIS
86
4.3. PROOFS
=
87
[(d~!J)",cf1~;L .. N-2,N)(d~~ 1 +NaJ]/INJ
Sym
[M))n d[MJ
( [MJ
[MJ
li[NJ
- [(d N 'a 1,{1, .. ·,N-2) dN-1- d,v )(J(N-1 N- 1)
.
(4.14-1)
Sym
Since the operand is restricted to the symmetric functions of the indices {1, ... , IV}, the second
term in the r.h.s. of eq. (4.144) is simplified as
[MJ)n
[MJ
[MJ
[MJ
l/[N)
- [( d,v
'ad1,{1, .. ,N-2)(dN-1 - d,v )(J(N-1 N- 1)
Sym
= adiMJ
(d[MJ
l,{l,···,N-2}
N-1
- d!MJ)(K
N
N-1
N
- 1)(d!MJ)"I!NJ
N
Sym
- ad[MJ
(d!MJ)"(d!MJ - JMI)(r
- 1)/[NJ
1,{1, .. ·,N-2) N
N-1
N
\N-1 N
Sym
1
1, 1,
= ad ''!J
··,N-2)(d~~1- d~tj)((d~~1)"J(N-1 N- (d\~ 1 JJ")/[NJ
Sym
= ad[M)
1,[1, .. ·,N-2}
(d(M) - d[M))((d[MJ )"- (d[M))")/[Nj
N-1
N
N-1
N
Sym
- (d[.ll)
d[M)
) ((d[Mj )" (d[Mj)")
-a 1,{1, .. -,N-1)- 1,{1,. .. ,N-2,N )
N-1 N
[NJ
I
(4.1-15)
Sym.
Using the inducti1·e assumption (4.143), we simplify the first term in the r.h.s. of eq. (-1.1-1-1)
as
[NJ
n d[MJ
.
(cfMJ + N J]/
[( d[MJ)
N
' 1,{1,···,/\ -2,N}
t'l-1
a Sym
{1 ... N-2
J ' '
= [(d[MJ)n d[M)
N
,
l,{l,···,N-2,N} 1Sym
N)
, (d[MJ + N )
a
N-1
N~
I[NJ
Sym
+ d[MJ
l ,{ l ,···,N-2,N}
--a d[MJ
("(d[AIJ)"-(N-?)(cfMJ)")(d[M)
+Na)
l,{l,···,N -2} ~
J
•
N
N-1
[(d[MJ)" d[~l) J
N
1
A-1
I["')
Sym
,~
Sym
F1
[MJ n
+ ad1,{ 1, .. ,N-2,N) ( (d,v-1)
By inducti\'e assumption, we assume the formula for the case N- 1, i.e. ,
d[MJ
ll{l, .. ·,N-2,S} = al"i, (JIMI(JV- 2)- (JV[( d[.I!J)n
N
' 1,{1, .. ·,N-2,N} Sym
1,[J\ -2j n
2)(d~IJJ")/{1 , .. ·,N-2,N}
l,[NJ
'
[Mj
x
j;1
+a
N-2
L
((d\~~ 1 )"- (d}Mil")- a(N- 2)((d~~~ 1 )"- (d~ 11 l")
[MJ
+ ad1,{1,.-,N-2,N)((d,v_1)
I
n
[.1/j n
- (d,v ) )
Sym
n
[Mj
( f[MJ
'If )
((d\~~ 1 + (N- l)a)(d\~" 1 + Na)- (d~ ! +(IV- 1)a)(d~~1 + iVaJ)]
) I[NJ
Sym
[N)
Sym
I
I[N)
N-2
= adiMJ
(L (c/Ml)"- (N- 2)(d\;~tJJ") Sym
1,{1,. .. ,N-1} j ; J
d[Mj
1,{1, .. ·,N-2}
1
ym
--a d[M)
((d[Mj
+ Na)(·"-(d[MJ)n(N?)(d[Mj)")
1,{1 ,-··,N-2}
N-1
L
1
.
N
j;1
[NJ
= ([ (d,v ) 'd1,{1,.--,N-2,,\'} C .V-1 + l a +
S
Sym
(4.143)
N
[N)
I
.Y-?
\\'e decompose the l.h.s. of eq. (4.1-12b) into two terms as
[ (d[MJ)" d[MJ l
[M) n
(dJ\. ) )
1
)I
[NJ
Sym
[.11) n
+ ad1 ,{1 ,... ,,,._, N) ((d,y_
1)
~,
-
[.\f j
n
(d,y ) )
[NJ
I
Sym
·
(-1.1-16)
88
Note that we have use d t he 1.d en t't'
1 ) (4 ·142a) se,·eral times in t he calculation. From eqs. (4.14-1 )
_ (4. 146), we finally obtain
d[M] JIIN] = adiM]
(IIMI(N- 1) - (N[( d[Af])n
N
> l,INJ
Sym
l,IN-1] "
1)(d\~lj)") I[Nj
,
Sym
h. h · 110 thin but the second formula of Lemma 4. 24 ·
w JCNowJS we shallg start the proo f of p ropos1.t.JOn 4· 21 · We .have
the case for
. to separately
.
_ prove
.
k = N because of the difference of the definition of the ra1smg operato1 (4.u4).
Proposition 4. 25
b+INI]
[/2IN!(N)
• > N,[N]
= b+INI
(2![NI(JV) +
N,IN]
[/2INI(JV) > b+[Ni]
N,IN]
=
:f[(dn2,aliNJ ..
j=l
~(d1·N]~tliNJ
.
~
....,
bt~[~J (4.54b) and the
·a~INJ]
tIN]
· · CiN
Then from Proposition 4. 4 and Proposition 4. 14 fork = 1, i.e., eq. (4. 124), we obtain the
results.
We assume that Proposition 4. 21 is valid up to l, name ly, k = 1 + 1 and N = 1 + M.
Assuming the validity of Proposition 4. 21 for the case of k and N, we shall verify the case of
k + 1 and N + l:
iN+!]
+IN+l]
[J2IN+l](N + 1) bk+l,IN+l]
l
>
l
Sym
I I ( 2/[N+l](N + 1) + (k + 1) + a(k + 1)(N- k)) +
= { b:;~.r;+l]
L g1:t~n~;,~ } IIN+l] .
J<;(N+l]
jJj=k+2
Sym
(4. 150)
Since both sides of the above expression are invari ant under the permutations of indices
1, 2, ... , N + 1, it is sufficient to check the term with the factor a~IN+l] ... arr;"J+l]a~J·~tl on
the left. Thus we introduce the restriction a(IN+lJ, ar~V/lJ, ... , a~N+l] ~ 0, which we denote by
lo: 1::,'.i .N-o·
From the definition of the conserved operator
J~N] (N)
(4.137) and Lemma 4. 15 ,
we have
!~N+l](N + 1)
)
+ 0:1tiN] . '. O:Nt IN]d[N]
j
j=l
t
89
(4.147)
N)).
This formula can be straightforwardly verified from the definition of
second formula of Lemma 4. 16 :
4.3. PROOFS
t(N+l]b+[N+l] (d[N+lj + (k + 1) ) + b+IN+l]
aN+! k,INJ
N+l
a
k+l,IN]·
Then the l.h.s. of eq. (4 .150) is decomposed as
j=l
[N+l]
+[N+l]
[/2
(N + 1), bk+l,(N+l]
b+IN]
) + N) ) .
N,INJ (2I[N](N
l
{[f2[N+l](i'Y)
As a ground for inductive assumption, we need a proposition:
Proposition 4. 26
>
N+l
ll
(4.152)
IIN+l]
o:l::,'.i ,N-o Sym
t(N+l] b+IN+l] (diN+l] + (k + 1)a)] + [JIN+l](JV) b+IN+l.J]
aN+l k,INJ
N+l
2
> k+l,INJ
tiN+!Jb+IN+lJ (diN+l] + (k + 1)a)] +[(diN+!]? b+[N+ll] } I
+[(diN+lJ?
L 9k~jln~'5i)
IIMJ
,
'
'
Sym
(4.151)
+IN+l]
bk+l,IN+l]
al[NJ ... a~[N](2d1N] + 1)
b+IMIJI IMJ = ((b+ IMJ 11MI(M)+l+a(M-l))+
[JIA!J(J\/)
> l,IMJ
Sym
l,[Mj 1
J<;[Mj
2
jJj=2
!~N+l](N) + (dr.:::i 1) 2 ,
k,[N]
'QN+ 1
N+l
N+ l
' k+ l ,[NJ
a:r~~:2~.~--.N ""O
VM:;:.: 1. (4 .148)
IIN+!J_
Sym
(4.1 53)
(4.15~a)
This is nothing but Proposit ion 4. 21 for l = 0, i. e., k = 1 and N = M ~Al]. The proof is as
follows. Using eq. (4.21c), we can rewnte the second conserYed operatm /2 (M) as
IN+l])2 tiN+!] t(N+ l]
t(N+l]diN+l]
= [( dN+J
, a,v+ 1 a2
· · · ak+l
1,{2,-··,k+l}
IN+lj 2 +[N+!JJI
[(dN+l ) 'bk+l,INJ
0
t[N+>I
-o
l
[N+!J,
.v+IJ
I011t.k+2 . . ,,v""O ISym
(4.15~b)
IIN+l]
Sym
l.k+2.···,N
t[N+l]~t .. t[N+lJ]diN+lJ
(diN+lJ
+ (k + 1)a)
[(d[N+l])2
N+l
, Q:t
u.2
Q:k+!
l,{2 ,···,k+l}
1
iE{l,k+2.-
·,N)
[N+l]
I
ot(S+IJ
l.k+2,···,N
...... o
ISym
.
(4.15-lc)
90
4.3. PROOFS
91
at[N+tJ . .. t[N+IJ t[N+!J]d[N+!J
{ [( d[N+!J)2
N+l
' 2
ak+l aN+ l
1,{2,-··,k+I,N+l)
Thus the r.h.s. of eq. (4.153) becomes
[It+tJ(N + 1), b;l~.i~~!J]
+atiN+!J ... at1N+Ilat1N+IJ [(d[N+!J)2 d[N+!J
]}I
2
k+l
N+l
N+l
' 1,{2,-··,k+I,N+!)
[N+!J(JV) , aN+l
t[N+!Jb+[N+!J]
+ (k + 1)a)
= { [/2
k,[Nj (iN+lj
N+l
t[N+!J t[N+tl
t[N+Ijd[N+!J
[!IN+Ij(N) d[~:il]
+aN+I a 2
· · · ak+I
t,{2,-··,k+I) 2
' '
t[N+!Jd[N+!J
2 t[N+!J t[N+!J
I
+[(d\::i ) , aN+ I a 2
· · · ak+I
t,{2,-··,k+t,N+t)
a~[N+!J · · · al~t!Jau•:t!J{ ( 2d\::il +
l
+
I
+a
"'
[(d[N+!J)2
t[N+Ij a2I . . . at{N+Ii]
d{N+!J
. ) (d{N+!J
+ (k + 1)a)}
~
N+l
, a;
k+I
1,{2,-··,k+t
•
iE{l ,k+2, ··,N)
[N+Ij
t{.V+II
0 !,k+2.···,N
-o ISym
2
+a . .
2
_
-
1
, aN+ !
k,[N]
(d\::il + (k + 1)a) I tiN+iJ
K;N+IJ(jN+I)d\'~;, 1.1 ,k+ I ,N+l)
k+l
+ad\~;,Ij,k+I) (~(d!N+Ij)2k(d!~:t'IJ2) } Ia:r::,'i
I[N+IJ
,N-o Sym .
(4. 159)
Using eq. (4.138b), we get the following expression from the fourth term:
I[N+!J
.
Io;~Z:z'.~ ... ,v"'O ISym
(4.156)
o,,k+2.···.N""O Sym
-a
L
a~[N+tJ. · · al~t 1 a~~i 11
iE{I ,k+2, .. ,N)
The above commutator allows the use of inductive assumption. Thus we have
t[N+!Jb+[N+!J] (d[N+!J + (k + 1)a)
[JIN+IJ(JV)
2
'QN+ l
k,[N]
N+l
Iot~Z:z~.~
K;N+Id!~":i 1 )
"'
[(d[N+IJ)2 0 t[N+ !J t[N+tJ .. . ~t[N+ti]d[N+Ij
(d[N+ !J + (k + 1)a)
L_;
N+ 1
' z
Q2
u:k+l
l,{2 ,.··,k+ l}
I
iE{I,k+2, .. ,N)
{N+!J
{N+I)
oE~:2'.~ ....v"'o ISym
N+l
{Vj
at[N+ti[IINI(N) b+iNI JI ' I
N+l
2
' k, 1N 1 Sym O(NJ.....,o(N+l]
k)
i;tj
(4.155)
.
t[N+ tlb +[N+tl ] (diN +II+ (k + 1)a) I
L
1 + a2 (N -
(2K;N+t + d\::i 1K;,v+l +
>,JE{l,k+2,-··,N)
We shall calculate the above equation term by term. Using Corollary 4. 23 , we can rewrite the
commutator of the first term of the r.h.s. of eq. (4.155) as
[/ [N+ !J( rlf)
L
iE{I ,k+2, ··,N)
I[N+!J
t(N+IJ
o,,J::+'2.· ··.N""'o Sym
[N+ I) + d[N+!JJ(
J(
( J(i N+ldN+I
N+l
iN+I + iN+!+ a+ a
{N+!J
~
"'
J'
]{
)
\j N+ l iN+ l
jE{l,k+2, .. ,N)
j¥i
. . N""o ISym
t[N+Ij · · · ak+I
t[N+lj aN+I
t[N+!Jd[N+!J
+ k + ak(N- k))
1,{2,-.. ,k+t,N+t) (2I[N+!J(JV)
I
{a 2
t[N+!J
t[N+!J t[N+!J d[N+Ij
(Nd[N+Ij- I [N+!J (N))
+2aa2
N+t
1
· · · ak+l aN+I 1,{2,-.. ,k+t)
[N+!J
t[N+!J
"'
{N+!J
{N+!J}
+ aN+l
~
9N -I,J\{N+ l)nk+2 ,J I,!IN+ll -O ISym .
J<;[kju{N +I}
i.k+2 ......v
N+IEJ,IJI=k+2
(4.160)
(4.157)
Assembling eqs. (4.157) - (4.160) and doing some calculation with the help of Lemma 4. 24
and the definition of null operators (4.59), we obtain
+[N+!J
[/2[N+!J (N + 1) , bk+t,[N+!J
lla:r::,'.i .
I[N+!J
.v-0 Sym
t[N+ IJaN+I
t[N+!Jd[N+!J
+ 1) + (k + 1) + a(k + 1)(/V- k))
1,{2,-.. ,k+I,N+l} (2I[N+Ij(N
I
{ O<zt[N+!J ... ak+l
[N+!J
"'
•IN+tJ IN+tJ }I
+
L...,
gk +2,J n k+2, J
t[N+ll
...... o S m
I
0
J ~[N+l)
t.k+2.···,N
y
N +IEJ,I JI=k+2
{ b;fr+!J ( 2l[N+!J(N + 1) + (k + 1) + a(k + 1)(/V - k))
By usi ng eqs. (4.138a) and (4.142b), the third term is cast into
t{S+!J
[(d[N+!J)2
N+l
'Q2
t{1\'+l) 0 t[.V+!Jd[N+!J
... Qk+l
N+l
1,{2,···,k+l,N+l}
l
+
[N+!J
t(S+i]
Io 1 .k+'1.· ·.N ..... oISym
"'
~
J c;:[N+ l ]
[N+I) [N+!J }
gk+2 ,Jnk+2,J
[N+!J
s
'
1
I
I0 t[.v+_,
l.k+-'.· .,,v"'O ym
lll=k+2
which prO\·es Proposition 4. 21 .
(4.161)
CHAPTER .J. ORTHOGONAL BASIS
92
4.3.5
4.3. PROOFS
93
Normalization, Triangularity and Expansion
Defining the Young tableaux from the sets of integers that appear in the r h s of the abO\·e
· · ·
expressiOn as exponents by
We shall prO\·e the last two properties of the Hi-Jack polynomials (4.50c) and (4.50d) and
Propositions 4. 7, 4. 8 and 4. 10 . Here again we implicitly assume that the Dunk! operators
depend only on N variables x 1, • • · , XN - The restriction symbol without explicit indication of
indices also means the restriction of the operand to symmetric functions of indices 1, 2, · · ·, N.
First we shall prol'e the following proposition.
Proposition 4. 27 Operation of symmetric polynomials of Dunk! operators {al, at· · ·, a~}
on any symmetric polynomials of N variables {x 1 , x 2 , · · · , XN} yields symmetric polynomials of
the N variables {x 1 , x2, · · ·, XN}.
Let us denote an arbitrary polynomial of x 1 , x 2, · · ·, XN by P(xt , x2, · · ·, XN ). Acting the Dunk!
operator
on P(x 1 , x2 , · · · , XN ), we hal'e
aJ
A(j±I)
Pt .... ,Aj ± 1, ... ,AN},
A(i -k,j+k- 1)
{AJ, ... 'A;- k, ... 'Aj + k- 1, ... ' AN} ,
{AI,·· · ,Aj- k, ·· · ,A;+ k -1,· · · ,\v },
A(j-k,i+k-1)
wh:re the integers in the r.h.s. are regarded to be arranged in the non-increasing order we
nottce the followmg relatiOns:
'
(-U6-l)
From eqs; (4.162), (4.163) and (4.164), we have the following result as a special case of PropoSition 4. _7 :
Coro lla r y 4. 28 Operation of the monomial symmetric polynomial whose arguments are Dunk!
operators on 1 yzelds a symmetric polynomial of the following expansion,
(4.162)
It is obvious that the first and the second terms of the r.h.s. of eq . (4.162) are polynomials of
x 1 , x2 ,. ·., x,v. Since the difference of polynomials P(x 1 , x2 , · · ·, x,v) - P(x; H xi) has a zero
at x; =xi, the third term is also a polynomial of x 1 , x 2 , · · ·, XN- Proposition 4. 27 follows from
this property. We note that the orders of the second and the third terms, which are the terms
with the coefficient 1/2w, are less than that of the first term by two in the r.h.s. of the abol'e
expression.
As a basis of symmetric polynomials, we employ the monomial symmetric polynomials
[50 , 69] defined by
(4.163)
u: distinct
permutation
where A is a Young tableau (4.15). By definition, monomial symmetric polynomials are symmetric polynomials with respect to any exchange of indices 1, 2, · · ·, N . Calculating the action
of the Dunk! operator a~(j) on the monomial x~(qx~( 2 J · • · x~('N)• we have
mA(a!,nt · · · ,o;v)
·1
=
mA(x 1,x2, · · · ,xN)
L
+
d
~9 and I~I<IAI
and I~I=IAI (mod2)
(4 .165)
where an unspecified coefficient (1/2w)OAI-I~D/ 2 uA~(a) is an integer coefficient polynomial of a
and 1/2w. In the above expansion, increasing the order of 1/2w by one causes decreasing of the
wezght of the symmetrized monomial by two.
Next we shall consider the action of d; operator on the monomial symmetric polynomials
of al, a~,···, a~v- We shall consider the case where the length of the Young tableau A, 1 is less
than or equal toN , l::::; N, i.e. , A= {>. 1 ~ A2 ~ ···~At> 0}. From the monomial symmetric
polynomial mA(al, a~,···, a\-), we single out a monomial (a~(l))A' (a~( 2 ))A' . . . (a~(t)lA,_ Because
of eq. (4.21d), we have
d;(a~ 1 !))A'(a~ 12 ))A' · · · (a~(l))A'
· 1 = [d;,
(a~(l))A'(a~ 12 /' ·· · (a~(l})A'J/
. 1.
(-1.166)
Sym
From eq. ( -l.2lc) , we can easily ,·erify
[d;, aj] =a! {oiJ(1 +a£ K;k)- a(1- o;1 )K;j }·
k=l
k;ii
Using the abo,·e formula, we get the following expressions:
[d;,
=
(a~lt/' (a~(2))A' ... (a~(l})A'J/
{ ( Ah
Sym
+ (.Y- l)a) (a~c 1 /' (a~12/' · · · (a~u/'
(-1.167)
4.3. PROOFS
95
After summing over the distinct
t .
and the fourth terms of (
)
~ermu atwn a E S,v , the terms that come from the third
4 168
a replaced by a'= a(jh) .yiel~s cance out and vanish, for the fourth term with the permutation
h
- a(atu'(l) )"'.
v
(4.171)
which is the bsummand in the thi r d t erm 0 f eq. (4 .168a ) wtth
· negati,·e sign. Similar cancellation
a 1so occurs etween the second term in the bracket of the first term of eq. (4.168a) and the
first term of eq. (4.168b) with a replaced by a'= a(kh) 1 + 1 < k < N ·1
· h 1
is that of eq. (4.168a):
,
'" lere am t e r. l.S.
h
( 4.168b)
e
Note that the summation
2.::: , e < s means zero, which occurs in the third and fourth
term of
k=s
eq. ( 4.168a) when >.h = >.1 or h = 1, l. From the above calculation, we get the followin g result.
Proposition 4. 29 No higher order monomial in the dominance order is generated by the
action of d; operator on a monomial (aj)-'•<•l(at)-'•(21 · · · (a~v)"•<Nl · 1, where>.
Pt ?:: >-2 ?::
· · · ?:: AN ?:: 0} and a E S N, and the coefficients of the monomials are integer coefficient
polynomials of a. The weight of the Young tableaux of the monomials are the same as that of
the original monomial.
=
This property causes the triangularity of the Hi-Jack polynomials (4.50c) . From the definition
of the raising operator of the Hi-Jack polynom ial and t he above proposition, we notice the weak
form of the third requ irement of Proposition 4. 3 .
a(a~,(ll' · · v. · · (a~·ul'(a~'(k))"•
-a(a~(ll'(a~c 2 l' . .. (a~(l))"'·
the commutator [d;, rn,~(al, a1, · ·· ,a~) Jl
is >.h where i =a( h).
S.
th H. J k 1
Sym
mce e 1- ac · po ynomial is symmetric with respect to the exchange of indices 1 . . . N
we have only to calculate the coefficient of (alJ"• ·· · (a/)"'· In the following calculation :ve shall
omit all the monomials except for (all"' · · · (aJ)"', namely the monomial with identit~ permutatiOn .. Any lower order monomial and the same order monomial with different permutation
are om1tted m the expression. However, we implicitly sum up over the distinct permutations to
use the above cancellation. To know the coefficient of the monomial of interest that is yielded
from b(rn,~(al, ··· ,a~) · 1, we have only to do the followin g calculation using eq. (4.168):
dl,{l, .,q(all"' ···(a/)"'·
+ la(a1)"' · · · (aJ)"'} · 1
Sym
(bt)-'N(bt_l)-'N-1-,\N ... (b"t)-'1-,\2 ·1
L
v,~~(a)rn~(al, ···,a~) ·1,
=dt,{l,
,2}{ (>-t + la)(a1)"' · · · (a[)"• + a(N -l)(aJ)"' ·· · (a/)"'+ a
(4.169)
-£
(all"' · ·· (a/)"'}· 1.
i=2
..\,<>.1
0
~-t$>.
where
v,~~(a)
(4.172)
Since there are N - l permutations a' that yield the monomial -a( at )"• (at )"' ... ( t )"'
2
through the commutator (4.168b ), we can cancel the second term ;ffhe fir~~ )bracket ~~(lihe
first term of eq. (4.168a). Thus the coefficient of the term (a~(l))"' ... (a~(l))"' coming from
=dl,{l, ,2} { [dt, (all"' · · · (a/)"'JI
Proposition 4. 30
=
is an unspecified integer coefficient polynomial of a.
Combining Corollary 4. 28 and Proposition 4. 30 , we can easily co nfirm the first formula of
Proposition 4. 9 symmetrized and hence the third requirement of Proposition 4. 3, i. e., eq.
( 4.50c). To ,·erify the second formula of Proposition 4. 9 , or equivalently, the fourth req uirem ent
(4.50d), we ha,·e to sho"·
(4.170)
Computation of """(a) needs consideration on the cancellation am ong the monomials in eq.
(4.168).
(4.173)
Summing o,·er the distinct permutation, we can cancel out the second and third term as has
been exp lain ed in eqs. (4.171) and (4. 172). Next, we operate d 2 on t he operand:
dl,{< . .. ,q(all"'
···(a[)>.,· 1
(>-t + la)dt,{l,
1
-a
L
i=l
).,>>.2
,3}{ (.-\ 2+
(1- 1)a)(a1J"• ···(a/)"'+ (N -l)(all"' · · · (a[)"•
N
(at)>.'(oJ)"'(a~)"'
(a/)"'+ a
L
i=3
Ai<A2
(all"'··· (a/)"'}· 1.
(4.17.J)
CHAPTER 4. ORTHOGO.VAL BASIS
96
Since we have already operated d 1 on the monomial symmetric polynomial m,~(al, · · · .' a:v),
summation 0 ,·er the distinct permutation is not anymore invariant under the permutation of
the indices 1, ... , N, but invariant under the permutation of indices 2, · · · , N. To cance l out
the second and fourth terms, we only need transpositions (2j), 3 ::; J :S N. The thrrd term ,
which is a monomial with the same Young tableau and a permutation (12), can not be canceled
out because of the break of invariance under the permutation involving the index 1. Howe,·er,
this monomial can not be changed to the monomial with identity permutation by operating d;,
4.3. PROOFS
97
is obvious that the Hi-Jack polynomials in the limit w --) oo are the Jack J)Olvnomials Thus
.
~~~
.
(4.179)
0
p<.l
which means the coefficients v,~p(a) in eq. (4 .38b) and v,~ 1 ,(a) in (4.169) are essentially same:
(4.180)
i :;:: 3. Thus, we have
dt,{l,··,l}(al).\ 1 · ··(a/).\'· 1 = (.!1 1 + la)(.i\z +(I- 1)a)dr ,{l, ··,J}(al).\
1
• • ·
(aJ).\
This proves Proposition 4. 8. Calculating thew--) oo limit of Proposition 4. 9, we ha,·e
1
•
1.
(4 .1 75)
j,~(x;w--) oo, 1/a)
Repeating analogous calculations, we get
dt,(l,···,l}(al).\ 1
• • •
(a/).11 · 1
= (.!1 1 + la)(.i- 2 + (l- 1)a) · · · (.!11 + a)(alJ.\
1
• • •
(aJ).\ 1 • 1.
which shows the top weight monomial symmetric functions in the expansion of the Hi-Jack polynomtal (4. 70a), or equtvalently, eq. (4.50c) form the Jack polynomial of the same Young tableau.
Smce the monomtal symmetric polynomials m,~(x) and the Jack polynomials J,~(x; 1/a) respectively form the bases of the symmetric polynomials, the lower weight monomial symmetric polynomials can be re\\-ritten by the lower weight Jack polynomials. Thus we confirm Proposition
4. 10.
{ (.!1 1 + la)(.i- 2 +(I- 1)a) · · · (.!11 + a)m.l+t' (a\,···, a:v)
Y.l+l'p(a)mp(al,
···,a~)}
J,~(x; 1/a),
(4.176)
bTm,~(al, · · · ,a~) · 1
L
w,~p(a)mp(x)
0
p<.l
Then the following expans ion follows from the above formula,
+
L
=
·1,
(4.177)
JJ.~>.+ tf
4.3.6
Orthogonality
wF.I
In our proof of Proposition 4. 11 , it is com·enient to use some results in previous sections.
where .il+ 11 = {.!1 1+ 1, · · ·, .!11+ 1} and Y.1+11P(a) is an unspecified integer coefficient polynomial
of a. We remark that 1 is a monomial symmetric polynomial with .>- = 0. Then from the
definition of the raising operator (4.54) and repeated use of eq. (4.177), we finally ,·erify eq.
(4.170):
Proposition 4 . 32 Operation of d; operator on a monomial of al 's, (at).l•tl) .. (a~).l•t.V>. 1,
where A E Y,v and u E SN yields the monomials of al 's of the form , (a()~•P> · · · (a:v)~·<·'> · 1,
Lemma 4. 31
The above assertion is a part of Proposition 4. 29 . We also use the following property.
D
where J.L
E YN,
J.L:SA and
T
= (i, h)u E SN,
1::;
h::; N.
Proposition 4. 33 The Hi-Jack polynomials are expanded into the monomial symmetric polynomials of al 's as
j,~(x; w, 1/a) =
Combining Corollary 4. 28 , Proposition 4. 30 and Lemma 4. 31 , we can confirm the th ird and
fourth requirements of Proposition 4. 3, eqs. (.J.50c) and (4.50d), and Propositions 4. 7 and -1.
9.
In the limit w --) oo, eq. ( 4.169) reduces to
(b;t).\'(b;t_y.v-1-.lv .. . wn-~~-.1,.
11
W->00
L
0
v,~P(a)m 1 ,(x 1 ,
· · ·,
xN)·
(4.178)
11<>.
As has been remarked in Sect ion -1.1 , the Dunk! operators for the Calogero model reduce to
corresponding Dunk! operators (-l.-19) for the Sutherland model in the limit w --) oo. Then it
L
v,~p(a)m 1 ,(a!, ·· · ,a~) · 1
(4.181a)
0
p<.l
J,~(al, · · ·,
whet·e
v,~,~(a) =
aL 1/a) · 1,
(4.18lb)
1 f= 0.
This result is sho'm in Propositions 4. 8 and 4. 30. We can readily sec that the above proposition a nd Corollary 4. 28 lead to the triangularity of the Hi-Jack polynomial (-1.50c). Here
we only use the triangularity (4.181a) and do not mind if the r.h.s. corresponds with the Jack
polynomial.
First, "·e shall pro,·e that t he Hi-Jack polynomials are the simultaneous eigenfunct ions of all
the commuting conserved operators {Inl1 ::; n ::; N}. By definition, the Hi-Jack polynomials
are the simultaneous eigenfunctions of the first two consen·ed operators, J, and fz. Due to
CHAPTER 4. ORTHOGO.YAL BASIS
98
Corollary 4. 28 and Propositions 4. 32 and 4. 33, operation of In on a Hi-Jack polynomial j,
4.3. PROOFS
99
The generating function of the elementary symmetric polynomials is given by
yields
N
E(t) ~ L en(,\)tn =
N
In· j,(x; w, 1/a)
:[(dtr :L v,~m 1 ,(al, ···,a~) · 1
l=l
n=O
D
~9
en(A) =en , n = l , 2, · · ·, JV,
0
~9
L
w~~m,(xi,· · · , xN) ·
(4.182)
o' I ~ I <IAI
Mnj,(x;w, 1/a) = EI(,\)In)A(x;w, 1/a) ,
hfnj,(x;w, 1/a) = E2(,\)Inj,(x;w, 1/a).
(4 .183a)
(4.183b)
Equations (4.183a), (4.183b) and (4.182) for In)A are respectively the same as eqs. (4.50a),
(4.50b) and (4.50c) for the Hi-Jack polynomial j,. We ha1·e proved in Section 4.3.1 the uniqueness of the Hi-Jack polynomial j, determined by the definition (4.50) . Therefore we conclude
that In]A must coincide with j, up to normalization. Thus we confirm that the Hi-Jack polynomials j, simultaneously diagonalize all the commuting conserved operators In, n = 1, · · · , !\".
Next, we shall verify that there is no degeneracy in the eigenvalues. The eigenvalue of the
n-th conserved operator for a Hi-Jack polynomial j, is denoted by En(,\):
Inj,(x; w, 1/a) = En(,\)j,(x; w, 1/a), n = 1, · · ·, N.
(4.184)
Both the conserved operator In and the Hi-Jack polynomial j, are polynomials of the coup lin g
parameter a, which can be readily recognized from the definition of the consen·ed operators
(4.24) and Proposition 4. 7. Thus the eigenvalue En(,\) is also a polynomial of a,
(4.185)
Considering the case a = 0, 11·hich corresponds to N free bosons confined in an external harmonic well, we can easily obtain the constant term e~0 l(,\) as
N
= :[ (,\k)".
(-U86)
k=I
It is clear that there is no degeneracy in the constant terms of the eigenvalues {e~,0 l(,\)ln =
1, ···,IV}. We shall confirm it by showing uniqueness of the solution of the follo11·ing algebraic
equations:
e~,0 )(,\) = Pn , n = 1, 2, · · ·, N.
(4.187)
\\'e define the elementary symmetric polynomial by
L
a:distinct
tk'
n=l
Since the n-th consen·ed operator commutes with the first and second consen·ed operator,
[II, In)= [hin) = 0, we have
permutations
(4.190)
are uniquely determined by the solution of the algebraic equation ,
~ N
n
1
E(t) - Lent = 0 => Ak = - - k = 1 2 · ·-, N, ti >_ t 2 >_ . . > t
0
p.~>.
en(AI,,\2,·· · ,,\,v)=
(4.189)
k==l
From the abm·e definition of the generating function, it is ob1·ious that the solution of the
algebraic equations,
:[v~~m~(al, ·-,a~) -1
e~o)(,\)
N
IT (1 + ,\kt).
Au(l)Au(2)···Au(n)·
(4.188)
' '
.-
N,
(-1191)
where tk is the k-th zero of the generating function in non-increasing order. For com·enience, we
take the maximum num_ber of the non-zero elements of the Young tableau as infinity, 1\" = 00 .
The loganthmic denvative of the generating function (4.189) yields
d
- logE(t) =
dl
00
,\
00
L 1 + Akt = L Pn(,\)(-t)''-I,
_k_
k=I
n=l
where Pn(,\) is the power sum of degree n. Paying attention to the fact that E(O) = 1, we can
calculate the definite integral of the above expression over t:
(-1)n-I
oo
--pn(,\)tn =log(:[ en(,\)tn),
n
n=O
00
00 (
1)n-I
en(,\)tn = exp(:[ ---Pn(,\)tn).
n= O
n=l
n
oo
L
(4.192a)
n=i
L
(H92b)
Comparing the terms with the same order oft, we can transform the power sums to the elementary symmetric polynomials and vice versa. We shou ld note that no higher order polynomial
appears in the transformation given by the above relations:
Pn
Prt(et,e2J· · ·,en),
e,l
e1l(Pt , P2 , ···,pn)·
For given constant parts of the eigem·alues, Pn, n = 1, 2, · · ·, 1\", we can uniquely identify the
corresponding parameters, en, n = 1, 2, · · ·, N, by the above transformation (4.192b). Then
the algebraic equations (4.187) arc cast into another forms (4.190). Thus the solution of the
a lgebraic equat ions (4.187) are uniquely gi1·en by the zeroes of the algebraic equation (4.191)
and hence we hm·e prm·ecl that there is no degeneracy in the constant terms of the eigem·alues.
Since the consen·ed operators In arc Hermitian operators concerning the inner product (4.16) ,
this pro,·es that the Hi-Jack polynomials are the orthogonal symmetric polynomials with respect
to the inner product.
From the explicit form of the weight function,
(4.193)
CHAPTER 4. ORTHOGO.V.4L BASIS
100
we conclude that the Hi-Jack polynomials are a multivariable generalization of the Hermite
polynomials [46]. We note that a proof based on the isomorphism between the Dunk! operators
for the Hi-Jack polynomials and those for the Jack po lynomials [62, 83] has been reported
[37, 84]. Our proof is a direct proof that does no t rely on the isomorphism. As has been
pointed out before, all the Dunk! operators for the Hi-J ack polynomials red uce to those for the
Jack polynomials in the limit , w --too. Thus our pro of contains a proof of orthogonali ty of the
Jack polynomials as a special case.
4.4
Summary
\\"e have studied t he orthogonal basis of the Calogero model. T wo cru cial observations are
the reasons why we ha,·e introduced t he Hi-Jack polynomials by Definition 4. 1 in a similar
way to a definition of the Jack polynomials, eqs. (4.38), which form t he ort hogonal basis of the
Sutherland model. One is the observat io n of the explicit forms of t he first se\·en orthogonal
symmetric polynomials (3.3-1) in Chapter 3 that indi cates a s imilarity between these polynomials a nd the Jack polynomials. The other is the common algebraic struct ure of the Calogero and
Sutherland models, which has been explicitly shown in Section 4. 1. \\·e have introd uced the
elementary deform ation of the Young tableau which generates all the Young tableaux 11 that
D
meet !1'5), from a Young tableau .A. Using the property of the deformation, we have confirmed
that our defin ition of t he Hi-Jack polynomials , Definition 4. 1 , uniquely specifies the Hi-J ack
polynomials. \Ye have proved that the fun ctions generated by the Rodrigues fo rmula [83] t hat
is an extension of t he Rod rigues formula for the Jack symmetric polynomials discovered by Lapointe a nd Vinet [42] satisfy the definition of the Hi-Jack polynomials. Our proof is based on the
algebraic relat ions among the Dunk! operators. In the consideration of their normalizations , we
have clarified t hat expansions of the Hi-Jack symmetric polynomials in terms of the monomial
symmetric polynomials have triangular forms , as is simi lar to the Jack sy mmetric polynomials. We have also confirmed that the Hi-Jack symmetric polynomials exhibits the integrali ty
corresponding to the weak form of the 1\Iacdonald-Stanley conj ect ure for the Jack symmetric
polynomials [41]. The Hi-J ack symmetric polynomials and the eigenfunctions for th e Hamiltonian that were algebraically constructed through the quantum Lax formulation in C hapter
3 a re related by the transformation between the Jack symmetric polynomials and the power
sum symmetric polynomials. \\"e have st udied on th e orthogonality of the Hi-Jack polynomials.
The orthogonal basis provides a ,·ery useful tool for th e study of physical quantities in quantum
theory. The orthogonality of t he Hi-Jack sy mmetric polynomials is expected to be importan t
in t he exact calculation of the thermodynamic quantities such as the Green function s and t he
correlation fun ctions, as has been clone for the Sutherland model using t he propert ies of t he
J ack polynomials [29, 30, 32, 48]. Orthogonality of the Jack polynomials is prO\·ed by showing
that all the co mmu t ing conse n ·ed operators of the Sutherla nd model {Idk = 1, 2, · · ·, N} arc
simul taneously cliagonalizecl by the Jack pol ynomials [49 , 50]. Co ns id ering the correspo nd ence
between the Calogero model and the Sutherland model , we can expect that all the conserved
operators of the Calogero model {hik = 1, 2, · · · , N} are also diagonalizecl by the Hi-Jack polynomials. This expec tat ion has been verified. We ha,·e verified that the Hi-Jack polynomial on
which the n-th co nserved operator operates, I,. jA, also sat isfies the definition of th e Hi-Jack
polynomial except for t he normalizat ion condition. This means that In)A coin cid es with the
Hi-J ack polynomial ]A up to a scalar factor. Therefore the Hi-Jack polynomials are the simul-
4.4. SUM.\IARY
101
taneous eigenfunctions of all the consen·ed operators. \\'e ha,·e calculated the eigem·alues of
h
h
·
a ll the conserved operators fa tl
.
r 1e case w en t e couphng parameter a is zero and ha,·e found
ouththere IS no degeneracy. Thus we have prO\·ed the orthogonalit\· of the Hi-Jack polynomials
:vit respect to the mner product (4. 76). From the explicit form ~f the weight function of the
mn er produ ct, we have confirmed that the Hi-Jack polynomials which we ha,·e found should be
regarded as a multivanable generalization of the Hermite polynomials.
--- ~--------
Chapter 5
Summary and Concluding Remarks
In this thesis , we have studied the quantum Calogero model in an algebraic fashion. In the
classical theory, the Calogero model is known to be a completely integrable system. The model
has a Lax formulation, which has been developed as a powerful tool for various completely
integrable systems. For the classical Calogero model, the Lax formu lation not only shows its
integrability, but also gives a way to solve the initial value problem of the model. In this sense,
the Calogero model is a special model even in the classical integrable systems, because the
notion of integrability just asserts the existence of the canonical transformation to the actionangle variables. However, because of the non-commutativity between the canonical conjugate
variables, the merits of the Lax formulation for the model seemed to be completely destroyed
in the quantum theory. This difficulty has motivated us to find a powerful method to studv
the integrability and the underlying symmetry of the quantum Calogero model. We hav~
also wanted to find out an elegant way to construct the eigenfunctions. Though the energy
eigem·alue problem was solved by Calogero in 1971 [19), an algebraic construction of the energy
eigenfunctions has not been completed. 'v\'e have pursued more detailed information on the
eigenfunct ions of the quantum Calogero model. Quantum integrability means that we can
identify, in principle, all the quantum number of the system by diagonalizing all the mutually
commuting consen·ed operators. But just proving it possible is far different from doing it in
practice. T hus we have aimed at giving a method to construct the simultaneous eigenfunctions
of the commuting conserved operators and hence to identify the orthogonal basis of the quantum
Calogero model.
In Chapter 2, "·e ha,·e developed the quantum Lax formulation of the Calogero model , "·hich
is a natural generalization of the Lax formu lation for the classical one, introduced the Dunk!
operator formulation and im·estigated the algebraic structure of the quantum Calogero model
in the framework of the quantum Lax formulation and the Dunk! operator formulation. From
the Lax equation for the classical model, we ha,·e obtained the Lax equation for the quantum
Calogero Hamiltonian whose J\I-matrix satisfies the sum-to-zero condition. The fact enables
us to construct a set of the conserved operators of the quantum model , as was the case wirh
the quantum Calogero-l\ loser model [34, 74 , 75, 88, 89). To show the quantum integrability of
the Calogero mode l, we have considered a construction of commuting consen·ed operators, eqs.
(2.12) a nd (2.20). By use of the explicit forms of the first two conserved operators, !1 and
h (2. 13) , or j 1 and j 2 (2.2 1), we have obtained the first two of the generalized Lax equations
for the commuting conserved operators (2.16) and ha,·e conjectured the general form (2.22).
However t he generalized Lax equations for the commuting consen·ed operators are not com103
104
CHAPTER 5. SU.\IMARY A. !YD CONCLUDING REMARKS
patible with the recursive construction. To study the recursive construction of the generalized
Lax equations, we ha,·e in troduced a series of operators that are made by summation of all
the elements of the \\'eyl ordered product of L+- and L-- matrices (2.29). From the explicit
form s of the first few conserved operators (2.26) and power-sum creat ion-annihilation operators
(2.30), we have obtained the corresponding generalized Lax equations , eqs. (2.28), (2.32) and
(2.33). From these first few generalized Lax equations, we have found out the recursion formu lae, eqs. (2.36) and (2.53). Using the recursion formulae, we have recursn·ely constructed the
generalized Lax equations (2.52). The generalized Lax equations prove t he mutual commutativity of the power-sum creat ion-ann ihil at ion operators, which ha,·e played an Important role
in the algebraic construction of the energy eigenfunctions [76, 77] in Chapter 3. Defining th_e
w <•>-operators by eq. (2.60), we have pro,·ed t hat the generalized Lax equatiOns yields the IIal;ebra as a com mutato r algebra amon g the IV~'> -operators. \\'e ha,·e studi ed correspondences
between the quantum Lax formul ation [76-78, 80] and the Dunk! operator formulation [27] that
was introduced by Polychronakos into the problems on the inverse-square interaction models
[62]. The Dunk! operator formulation provides us a simple way of constructing the commutin g
conserved operators of the Calogero model (2 .69). We have observed that the restriction of the
operand to the wave functions of the identical particles enables us to translate the results in one
of the theories into those in the other. To be concrete, we have related arbitrary operators made
from the two matrices , L + and L -, and their commutator algebras with those in the Dunk!
operator formulation , eqs. (2. 73) and (2. 76). A method to directly obtain th e M{;,-matrices has
also been given as eq. (2.97). Mutual commutativity of the conserved operators In has been
proved. The simultaneous eigenfunctions of these mutually commuting conserved operators
have been studied in Chapters 3 and 4.
In Chapter 3, we have studied an algebraic construction of all the eigenfunctions of the
Calogero Hamiltonian with the help of the quantum Lax formulation [76-78]. Our approach
is based on Perelomov's idea on an algebraic treatment of the eigenfunctions of the Calogero
model. By the factor ization of the Hamiltonian , we have obtained the ground state wa,·e
function. Using the power-sum creation operators which has been obtained in Chapter 2, \\·e
have obtained an algebraic method to construct the eigenfunctions of the Calogero model.
From the number of independent eigenfunctions , we have confirmed that the eigenfunctions
form the basis of the Hilbert space of the Calogero model. We have also reproduced the resul t
for the original Calogero model by fixing the center of mass at the coordinate origin. Thus we
have completed Perelomo,·'s dream of the algebraic construction of the eigenfunctions of the
Calogero model. Another approach to the algebraic construction of the eigenfunction of the
Calogero Hamiltonian that uses the Dunk! operators was also reported [17, 18], which has been
briefly summarized in C hapter 4. \\'e have also considered a const ruct ion of the orthogonal
basis of the Calogero model by diagonalizing mutually commuting conserved operators. \\"e
have directly diagonalized the first nontri,·ial conserved operator 12 using th e eigenfunctions
with weights up to 6. The results indicate a general formula for the eigenvalue of 12. In
addition, we have presented explicit expressions of the first seven of the unidentified orthogonal
sy mmetric polynomials associated wi t h the Calogero model, which have been identified as th e
Hi-Jack polynomials in Chapter 4. From the explicit form of the weight function of the innerprodu ct, we ha,·e concluded that the orthogonal symmetric polynomial should be regarded as
a multivariable generalization of th e Hermite polynomial. From the eigem·alue formula for the
commuting consen·ed operator / 2 and the expansion of the explicit form s with respect to the
monomial sy mmet ric function, we ha,·e obsen·ed a similarity between the Jack polynomials and
105
the seven orthogonal svmmetric pol ·
· 1 Th
1
·
"
.
) nOima s.
e resu ts have given an im portant sugrrestion
. d h d fi . .
t O\\ar t e e m t10n of the HI-Jack polynomials.
o
In Chafter 4 • we have studied the orthogonal basis of the Calogero model further and also
ave competed the construction of the orthogonal basis in an algebraic fashion [79 83 8-1 86]
Based on the fact that the Calogero model has a set of mu t ually commuti ng co~ser,ved op~
erators [62, 76-78, 80], we have tned to construct the simultaneous eigenfunctions for all the
conserved operators of the Calogero model that must form t he orthogonal basis of the mode l
Smce the Calogero and the Sutherland models share the common algebraic structu re, it is nat~
ural to mtroduce the HI-J ack symmetric polynomials in a similar way to a definition of the
Jack polynomials [83, 8-1]. The resu lts in Chapter 3 also s upport t his expectation. \\"e ha,·e
mtroduced the element~ry deformation of the Young tableau which generates a ll the Young
h
tableaux Jl. that meet Jl.<::,>. from a Young tableau >.. Using the property of the deformation we
have confirmed that our definition of the Hi-Jack polynomials, Definition 4. 1 , uniqueh· s~ec
Ifies th e HI-Jack poly nomials._ \\'e have proved that the fun ctions generated by th e Rodrigues
formula [83] satisfy the defimt10n of the Hi-Jack polynomia ls . This resu lt is an extension of
the Rodngues formula for the J ac k symmetric polynomials discovered by Lapointe and Vinet
[42]. Our proof is based on the algebraic relations among the Dunk! operators. In the conSideratiOn of their normali zations, we have clarified that expansions of the Hi-J ack symmetric
polynomials 111 terms of the monomial symmetric polynomials have triangularity, as is similar to the Jack symmetric poly nomia ls. We have also con firm ed that the Hi-Jack svmmetric
poly nomials exhibits the integrality corresponding to the weak form of the l\lacdonald-Stanley
conJ ecture for the Jack symmetric polynomials [41]. The Hi-Jack symmet ric polynomials a nd
the eigenfunctions for the Hamil tonian that was algebraically constructed through the quantum
Lax formulation in Chapter 3 are related by the transformation between the Jack s\·mmetric
polynomials and the power sum symmetric polynomials. We have confirmed the orth-ogonality
of the Hi-Jack poly nomials. The orthogonal basis provides a very useful tool for the study of
physical quantities in quantum theory. The orthogonality of the Hi-J ack symmetric polynomials is expected to be important in th e exact calculation of the th ermody namic quantities such
as the Green functions and the correlation functions , as has been done for th e Sutherland model
using the properties of the Jack polynomials [29-32, 48]. Orthogonality of th e Jack polynomials
is proved by showing that all the commuting conserved operators of the Sutherland model, Ik ,
k = 1, 2, · · · , N, are simultaneously diagonalized by the Jack poly nomials [49, 50]. Considering
the correspondence betwee n the Calogero model and the Sutherland model, we can expect that
all the conserved operators of the Calogero model , h , k = 1, 2, · · · , N , are also diagonalized by
the Hi-Jack polynomials . This expectation has been w rifi ed. \Ve have prO\·ed that the Hi-J ack
polynomial on which the n-th consen·ed operator operates, Inj~, also satisfies the definition of
the Hi-Jack polynomi a l except for the normalization condi t ion. This means that Inj~ coincides
with th e Hi-Jack polynomi al j~ up to a scalar factor a nd therefore the Hi-J ack polynomials are
the simultaneous eigenfunct ions of all the consen·ed operators. \\'e ha,·e calcu lated the eigen,·alues of a ll the conserved operators for the case when the coupling parameter a is zero and
have found out there is no degeneracy. Thus we ha,·e prO\·ed the orthogonality of t he Hi-J ac k
polynomials with respect to the inner product (4.76). From the explicit form of the weigh t
function of the inner product, we ha,·e concluded that the Hi-Jack polynomial is a multi,·ariable generalization of the Hermite polynomial. .-\ccording to recent preprints [11 , 12 . 26], the
generalized Hermite polynomials were also introduced by Lasselle [46] and by i\lacdonald in a n
106
CHAPTER 5. SUMMARY A.VD CONCLUDING REMARKS
unpublished manuscript. They defined the generalized Hermite polynomials as a deformation
of known orthogonal symmetric polynomials with respect to the inner product (4. 76) with the
parameter a fixed at 1/2. On the other hand, our definition specifies the Hi-Jack polynomials,
or in other words, the generalized Hermite polynomials, as the simultaneous eigenfunctions
for the commuting conserved operators of the Calogero model , which are natural objects for
physicists' interest.
Compactly summarizing, we have developed the formulation for an algebraic study of the
quantum Calogero model. We have presented a way to construct the commuting conserved
operators, and have identified the underlying symmetry. We have given a simple method for the
algebraic construction of the energy eigenfunctions. We have also identified the simultaneous
eigenfunctions of all the commuting conserved operators and ha1·e presented their Rodrigues
formula. The simultaneous eigenfunction is a one-parameter deformation of the well-known Jack
polynomial which we call Hi-Jack polynomial. Proving their orthogonality, we have identified
that the Hi-Jack polynomial as a multi,·ariable generalization of the Hermite polynomial. Thus
we have succeeded in clarifying the fundamental properties on the quantum Calogero model
from the viewpoint of the quantum integrable systems.
'vVe believe that the thesis has opened the gate toward the exact calculations of various
corre lation functions of the Calogero model. As we have mentioned before, the knowledge on
the Jack polynomials [36, 50, 69) enabled the exact calculations of correlation functions of the
Sutherland model [29-32, 48). The correlation functions exhibit an interesting connection with
the exclusion statistics [33], which was first recognized by the asymptotic Bethe ansatz [71] of
the Sutherland model [15, 91, 92]. Similar connections between the exclusion statistics and the
quantum Calogero model were also reported [53, 54, 81, 82, 85]. Thus we expect similar relationships between the exclusion statistics and the correlation functions of the Calogero model.
Some progresses related to our results were also reported recently [11, 12, 26). Multi variable
generalizations of the classical Laguerre and Jacobi polynomials, which form the orthogonal
basis of Calogero models associated with root lattices other than A,v_ 1 [59], were studied.
Their non -symmetric extensions, which describe the spin or multi-component generalizations
of Calogero models, were also reported. Further investigations on these orthogonal polynomials must be important for the study of Green functions and cor relation functions of Calogero
models, which was done for the Sutherland model with the he lp of t he properties of the Jack
polynom ials [29, 30, 32, 48]. The quantum La,, fo rmulation and the Dunk! operator formulation
for the Calogero and Sutherland models revealed their IV -symmetry and the Yangian symmetry structures [13, 14, 34, 35, 77, 78). It is interesting to study the orthogonal polynomials from
the viewpoint of representation theory of such symmetries. As examp les of such studies, we
should note that the Jack polynomials were identified with the singular vectors of the Virasoro
and ll'walgebras [10, 51). The i\lacdonald polynomials [50) and their generalizat ions, which
are q-deformations of the orthogonal symmetric polynomials, are also interesting topics. The
i\lacdonald polynomials are associated with t he discretization, or in other words, the relativistic generalization of the Sutherland mode l [63). The Rodrigues formula for the Macdonald
polynomials was given [43-45, 55, 56). Further studies on the cont inuous Hahn po lynom ials [23)
which are associated with the relati1·istic or disc retized Calogero model [22) and more generalized q-deformed orthogonal polynomials such as BC,v-Askey-Wilson polynom ials [39] related
to the discretized Calogero model associated with root lattices of BC,v-type [24, 25] are interesting. Thus we believe that the field of t he quantum integrable systems with inverse-square
interactions will continue to produce lots of attractive open problems.
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Algebraic Study on the Quantum Calogero Model

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